INTKIIMKDIATK    AHI-'.iM 


SCHOOLS  AM)  A"  ADKM1K8 


1  T  I-;  1 »    S  T  .V  T  E  S 


I  I 


[)  N"  AN,  Jfc., 

9   n-:i>  .»sa,  :.A 


XfcW     YORK: 

THK  TK^^,  ><!   CO. 

'»0. 


IN  MEMORIAM 
FLOR1AN  CAJORI 


INTERMEDIATE   ARITHMETIC 


FOR  USE   IX   THE 


SCHOOLS  AND  ACADEMIES 


UNITED      STATES. 


DETnns  cr.q-XA\.  -Tr. 

PRINCIPAL   OF    THE    "3lEXVII,ii;    B0\3'   6*CIl"OuLJ    >KW"  ORLEANS,    LA. 


NEW  YORK: 
THE  TROW  &  SMITH  BOOK  MANUFACTURING  CO., 

46,  48,  50  GKEENE  STEEET. 
1869. 


Entered,  according  to  Act  of  Congress,  in  the  year  1839,  by 
DENNIS  CKONAN,  JR., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States 
for  the  Eastern  "District  of  Louisiana. 


PREFACE 


IN  submitting  ray  INTERMEDIATE  ARITHMETIC  to 
the  consideration  of  my  fellow-teachers,  I  take  this 
opportunity  to  make  a  few  remarks  in  regard  to  the 
several  characteristics  of  my  work.  It  has  been  my 
aim  to  lead  the  pupil  gradually  into  the  mysteries  of 
numbers  ;  from  the  beginning  I  have  tried  to  advance 
the  pupil  slowly,  step  by  step,  until  he  at  last  arrives 
at  those  intricate  problems  in  Denominate  Numbers. 
He  is  now  taught  to  reason  with  them.  The  prob- 
lems are  numerous,  and  cover  every  rule  in  Denomi- 
nate Numbers,  from  the  most  simple  to  the  most 
difficult.  In  the  perusal  of  this  work,  no  one  will 
fail  to  note  that  I  have  written  this  volume  in  the 
same  language  in  which  it  is  taught,  using  no  terms 
beyond  the  comprehension  of  any  pupil  fit  to  study 
this  difficult  branch.  My  explanations  are  plain  and 
simple,  and  will  be  readily  understood  by  any  pupil. 
I  would  respectfully  call  the  attention  of  my  fellow- 


PEEFACE. 

teachers  to  the  system  of  teaching  Fractions,  wliicl 
I  think  will  answer  all  purposes  for  which  it  was 
intended. 

Practice  in  U.  S.  Money,  Denominate  Numbers, 
and  Duodecimals,  are  treated  to  their  fullest  extent, 
and  I  am  satisfied  that  there  are  enough  of  examples 
in  this  little  volume  to  satisfy  any  teacher.  I  have 
taken  great  care  to  demonstrate  every  principle  and 
to  give  a  plain  analysis  of  the  methods  employed 
from  the  most  simple  to  the  most  intricate,  and  to  ex- 
plain the  reasons  for  every  rule.  Satisfied  that  a  trial 
of  this  volume  will  meet  with  success,  I  leave  it  to 
the  jury  of  my  fellow-teachers,  who  alone  are  to  judge 
of  its  merits,  and  if  it  meets  with  their  approval  my 
labors  will  have  their  reward. 


N.  ORLEANS,  July, 


D.  CRONAN,  JR., 

Principal  Bienville  School,  New  Orleans. 


CONTENTS. 


PAGE 

Definitions 9 

Notation 10 

Arabic  Notation 11 

Numeration 20 

Properties  of  the  9's 22 

Addition 24 

Subtraction 3Y 

Multiplication 49 

Division G8 

Short  Division , 71 

Long  Division 75  ' 

Division  by  Composite  Numbers 80 

To  Buy  and  Sell  by  the  Hundreds  and  Thousands,  &c 83 

Examples  in  Numeration,  Notation,  and  the  Four  Ground  Rules  88 

Questions  involving  Fractions 95 

Contractions    103 

Practice  in  United  States  Money 121 

1* 


CONTENTS. 

PAGE 

Addition  of  United  States  Money 134 

Subtraction  of  United  States  Money .- 137 

Multiplication  of  United  States  Money 140 

Division  of  United  States  Money 142 

Questions  by  Analysis 146 

Properties  of  Numbers 151 

Divisibility  of  Numbers 153 

Factoring 154 

Greatest  Common  Divisor 158 

Least  Common  Multiple 163 

Cancellation 170 

Fractions 172 

To  Keduce  a  Fraction  to  its  Lowest  Terms 174 

Addition  of  Fractions 183 

Subtraction  of  Fractions 194 

Multiplication  of  Fractions 206 

Division  of  Fractions 215 

Complex  Fractions , 220 

Addition  of  Complex  Fractions 221 

Subtraction  of  Complex  Fractions 222 

Multiplication  of  Complex  Fractions 223 

Division  of  Complex  Fractions 225 

Brackets 227 

Reciprocals 230 

Decimals 231 

Addition  of  Decimals ,  234 


CONTENTS.  7 

PAGB 

Subtraction  of  Decimals 2oo 

Multiplication  of  Decimals 237 

Division  of  Decimals 2 10 

Reduction  of  Fractions  to  Decimals 243 

Reduction  of  Decimals  to  Common  Fractions 247 

Repetends 252 

Denominate  Numbers 2G2 

English  Money 203 

Troy  Weight 205 

Apothecaries'  Weight 266 

Avoirdupois  Weight 207 

Cloth  Measure ^ 260 

Long  Measure 270 

Surveyors'  Measure 272 

Square  Measure 273 

Cubic  or  Solid  Measure 276 

Wine  Measure 278 

Beer  Measure 281 

Dry  Measure 282 

Circular  Measure 289 

Duodecimals 292 

Reduction  Descending 293 

Reduction  Ascending £93 

Reduction  of  Denominate  Fractions  to  Compound  Numbers  . .  304 

Reduction  of  Denominate  Numbers  to  Compound  Fractions. . .  309 

Reduction  of  Compound  Numbers  to  Decimals 314: 


8 


CONTENTS. 


Reduction  of  Denominate  Decimals  to  Compound  Numbers. . .  321 

Addition  of  Denominate  Numbers 326 

Subtraction  of  Denominate  Numbers 335 

Difference  of  Dates 345 

Multiplication  of  Compound  Numbers 347 

Division  of  Compound  Numbers 335 

Duodecimals 862 

"          Addition  of 363 

"         Subtraction  of 370 

"         Multiplication  of 374 

"          Division  of 378 

Percentage * 384 

Interest 388 

Foreign  Money,  Rates  of 389 


AEITHMETIO. 


CHAPTER    I. 

LESSON  I. 

DEFINITIONS. 

B .  ARITHMETIC  is  the  science  of  numbers. 
S.  A  figure  is  a  character  used  to  represent  a 
certain  number  of  objects. 

3.  An  object  is  anything  which  can  be  seen, 
heard,  or  imagined. 

4.  Numbers  are  either  concrete  or  abstract. 

5.  Concrete  numbers  are  used  to  designate  par- 
ticular objects :  as  3  feet,  4  inches. 

6.  Abstract  numbers  do  not  refer  to  any  object 
in  particular,  and  are  merely  numbers,  having  no 
reference  to  anything:  as  1,  2,  3,  4,  5,  6. 

7.  A  unit  is  the  first  figure  of  the  nine  digits, 
and  represents  but  one  object. 

8.  A  number  is  a  collection  of  units  arranged  in 
order,  according  to  their  values. 

0.  The  fundamental  principles,  or  the  four 
ground  rules  of  arithmetic  are,  Addition,  Subtrac- 
tion, Multiplication,  and  Division. 


19 


Il^TEIlMEDIATE   ARITHMETIC. 


10.  They  are  called  the  four  ground  rules,  be- 
cause all  problems,  however  difficult,  depend  upon 
one  or  several  of  them  for  thiir  solution. 

Quantity   is   applied  to    anything  which  can  be 
measured. 

11.  A  fractional  number  is  a  part  of  a   unit. 
An  integral  number  is  a  whole  number. 

\2.  A  decimal  fraction  represents  the  object  as 
divided  into  tenths,  hundredths,  etc.,  etc. 


NOTATION. 

13.  Notation  is  the  art  of  expressing  numbers 
by  means  of  figures,  and  in  some  instances  by  means 
of  letters. 

14.  The   latter  method   is   called   the   Roman 
method,  and   is   used  in  numbering   the  chapters, 
lessons,  etc.,  of  a  book.     It  is  also  used  to  mark  the 
value  of  bank-notes,  etc. 

15.  The  letters  on  the  dial  of  a  clock  are  of 
the  Roman  method.     It  is  used  for  a  great  many 
other  purposes. 

1 6.  The  first  method  is  called  the  Arabic  method, 
from  its  having  been  employed  by  the  Arabians. 

1 7.  The  ten  Roman  digits  are  formed  thus : 

I,  II,  III,   IV,   V,  VI,  VII,   VIII,   IX,   X. 

123456  7  8  9          10 

1  8.  The  Arabic  digits  are  supposed  to  have  been 
derived  from  the  following  symbols,  each  one  having 


NOTATION. 


11 


tho  necessary  number  of  straight  lines  to  indicate  its 
value,  thus : 

z 

and  the  f  J  which  represented  nothing,  having   no 

straight  line,  or  because  it  represented   something 
which  had  no  beginning  or  end. 


The  figure 
The  figure 
The  figure 
The  figure 
The  figure 
The  figure 
The  figure 
The  figure 
The  figure 
The  figure 
The  figure 


LESSOR  II. 

ARABIC    NOTATION. 

1  one      represents    1  one  unit. 

2  two     represents    2  two  units. 

3  three  represents    3  three  units. 

4  four    represents    4  four  units. 

5  five     represents    5  five  units. 
G  six      represents    G  six  units. 

7  seven  represents    7  seven  units. 

8  eight  represents    8  eight  units. 

9  nine    represents    9  nine  units. 
10  ten      represents  10  ten  units. 

0  represents  nothing. 


LESSON  III. 

1O.  Write   in  figures,  one,  three,  five,    seven, 
nine,  two,  four,  six,  eight,  ten,  naught. 


12  INTERMEDIATE    ARITHMETIC. 


LESSON   IY. 

SO.  To  represent  any  number  between  ten  and 
twenty,  place  the  figure  representing  the  number  of 
units  to  the  right  of  the  figure,  1  (one.) 

Thus,  to  represent  12  (twelve),  place  the  figure  2 
(two)  to  the  right  of  the  figure  1  (one) — 12. 

Write  in  figures,  eleven,  twelve,  thirteen,  four- 
teen, fifteen,  sixteen,  seventeen,  eighteen,  nineteen. 


LESSON  V. 

21.  To  write  in  figures  any  number  between 
twenty  and  twenty-nine,  inclusive,  write  the  figure 
representing  the  units  to  the  right  of  the  figure  2 
(two.) 

Thus,  to  write  the  number  twenty-one,  place  the 
the  figure  one  to  the  right  of  the  figure  two — 21. 

Twenty-one  represents  two  tens  and  one  unit. 

Write  in  figures,  twenty-one,  twenty-two,  twenty- 
three,  twenty-four,  twenty-five,  twenty-six,  twenty- 
seven,  twenty-eight,  twenty-nine. 


LESSON  VI. 

22.  To  write  in  figures  any  number  from  thirty 
to  thirty-nine,  inclusive,  write  the  figure  repre- 
senting the  units  to  the  right  of  the  figure  three. 

To  write  the  number  thirty-four,  write  the  figure 
four  to  the  right  of  the  figure  three  :  thus,  34. 


NOTATION.  1 3 

Thirty-four  represents  three  tens  and  four  units. 

Write  in  figures,  thirty,  thirty-one,  thirty-two, 
thirty-three,  thirty-four,  thirty-five,  thirty-six,  thirty- 
seven,  thirty-eight,  thirty-nine. 


LESSON  VII. 

S3.  To  write  in  figures  any  number  from  forty 
to  forty-nine,  inclusive,  write  the  figure  representing 
the  units  to  the  right  of  the  figure  four. 

24.  To  write  forty-five,  write  the  figure  five  to 
the  right  of  the  figure  four :  thus,  45. 

Forty-five  represents  four  tens  and  five  units. 

Write  in  figures,  forty,  forty-one,  forty-two,  forty- 
three,  forty-four,  forty-five,  forty-six,  forty-seven, 
forty-eight,  forty-nine. 


LESSOX  VIII. 

25.  To  write  in  figures  any  number  from  fifty 
to  fifty-nine,  inclusive,  place  the  figure  representing 
the  units  to  the  right  of  the  figure  five. 

To  write  fifty-four,  place  the  figure  four  to  the 
right  -of  the  figure  five,  thus  :  54. 

54  represents  five  tens  and  four  units. 

Write  in  figures,  fifty,  fifty-one,  fifty-two,  fifty- 
three,  fifty-four,  fifty-five,  fifty-six,  fifty-seven,  fifty- 
eight,  fifty-nine. 


14  INTERMEDIATE   ARITHMETIC. 


LESSOR  IX. 

3G.  To  write  in  figures  any  number  from  sixty 
to  sixty-nine,  inclusive,  place  the  figure  representing 
the  units  to  the  right  of  the  figure  six. 

To  write  sixty-eight,  place  the  figure  eight  to  the 
right  of  the  figure  six  :  thus,  68. 

Sixty-eight  represents  six  tens  and  eight  units. 

Write  in  figures,  sixty,  sixty-one,  sixty-two,  six- 
ty-three, sixty-four,  sixty-five,  sixty-six,  sixty-seven, 
sixty-eight,  sixty-nine. 


LESSON  X. 

ST.  To  write  in  figures  any  number  from  seven- 
ty to  seventy-nine,  inclusive,  place  the  figure  repre- 
senting the  units  to  the  right  of  the  figure  seven. 

To  write  seventy-nine,  place  the  figure  nine  to 
the  right  of  the  figure  seven:  thus,  79. 

Seventy-nine  represents  seven  tens  and  nine  units. 

Write  in  figures,  seventy,  seventy-one,  seventy- 
two,  seventy-three,  seventy-four,  seventy-five,  seven- 
ty-six, seventy-seven,  seventy-eight,  seventy-nine. 


LESSON  XI. 

28.  To  write  in  figures  any  number  from  eighty 
to  eighty-nine,  inclusive,  place  the  figure  represent- 
ing the  units  to  the  right  of  the  figure  eight. 


NOTATION.  15 

To  write  the  number  eighty-four,  place  the  four 
to  the  right  of  the  figure  eight :  thus,  84. 

Eighty-four  represents  eight  tens  and  four  units. 

Write  in  figures,  eighty,  eighty-one,  eight  y- 
two,  eighty-three,  eighty -four,  eighty-five,  eighty-six, 
eighty-seven,  eighty-eight,  eighty-nine. 


LESSON  XII. 

29.  To  write  in  figures  any  number  from  ninety 
to  ninety-nine,  inclusive,  place  the  figure  representing 
the  units  to  the  right  of  the  figure  nine. 

To  write  the  number  ninety-eight,  place  the  fig- 
ure eight  to  the  right  of  the  figure  nine  :  thus,  98. 

Ninety-eight  (98)  represents  9  teas  and  8  units. 


30.  UNITS  OF  THE  THIRD  ORDER,  OK  HUNDREDS. 

LESSON  XIII. 

354  is  read  three  hundred  and  fifty-four,  because 
it  is  composed  of  3  hundreds,  5  tens,  and  4  units. 

To  write  in  figures  any  number  from  one  hun- 
dred to  nine  hundred  and  ninety-nine :  first,  write 
the  figure  representing  the  hundreds;  to  the  right 
of  it  place  the  figure  representing  the  tens  ;  to  the 
right  of  the  tens  place  the  figure  representing  the 
units. 

To  write  eight-hundred  and  ninety-seven:  first, 


16  INTERMEDIATE   ARITHMETIC. 

write  the  figure  8,  which  represents  the  hundreds ; 
to  the  right  of  it  place  the  figure  9,  which  represents 
the  tens,  followed  by  the  figure  7.,  which  represents 
the  units,  and  the  number  formed  will  be  897. 

Write  in  figures  (and  give  the  reason)  the  fol- 
lowing : 

Eight  hundred  and  twenty-five. 

Two  hundred  and  forty-six. 

Six  hundred  and  thirty-four. 

One  hundred  and  thirty-six. 

Seven  hundred  and  forty-two. 

Five  hundred  and  twenty -two. 

Three  hundred  and  sixteen. 

Nine  hundred  and  forty-eight. 

Four  hundred  and  ninety-two. 

LESSON"  XIY. 

31.  The  figure  0  has  no  value,  and  is  only  used 
to  fill  omissions  in  counting. 

If  we  wish  to  write  in  figures  seven  hundred 
and  four,  we  see  immediately  that  the  figure  repre- 
senting the  tens  is  wanting  to  fill  the  space,  but  since 
we  do  not  require  tens,  we  place  the  figure  0  in  the 
place  which  should  be  occupied  by  the  tens  ;  as  it  has 
no  value,  it  serves  to  show  that  there  are  no  tens, 
and  also  to  form  one  more  space. 

The  figure  written  would  be  704,  which  is  read 
seven  hundreds,  no  tens,  and  four  units. 

Write  in  figures : 

One  hundred  and  sixty.          Ans.  160. 


NOTATION.  17 

In  the  preceding  we  have  one  hundred,  six  tens, 
and  no  units. 

Write  in  figures : 

Two  hundred  and  forty. 
Six  hundred  and  eight. 
Nine.Tmndred  and  sixty. 
Three  hundred  and  three. 
Four  hundred  and  five. 
Five  hundred  and  fifty. 
Seven  hundred  and  eighty. 
Eight  hundred  and  ten. 
One  hundred  and  one. 

LESSON  XV. 

32.  A  figure  is  valued  according  to  the  position 
it  occupies  in  connection  with  other  figures. 

Thus,  the  figure  7,  in  its  simple  value,  means 
nothing  more  than  seven  whole  units.  By  placing 
the  figure  five  (5)  to  the  right  of  it,  its  value  is  in- 
creased tenfold,  that  is,  it  is  made  to  occupy  the 
second  place  or  order,  and  consequently  it  comes 
under  the  denomination  of  tens.  By  the  placing  of 
the  figure  five  to  the  right  of  the  figure  7,  we  make 
a  number  ten  times  as  great  as  the  original  number. 
If  an  0  was  placed  in  the  position  occupied  by  the 
figure  five  in  the  above  example,  the  same  result 
would  be  obtained,  as  regards  the  figure  7.  If, 
after  the  number  75,  the  figure  nine  is  placed,  the 
seven  is  removed  from  the  tens  to  the  hundreds,  the 
five  from  the  units  to  the  tens.  The  number  thus 
2* 


18 


INTEEMEDTATE    AEITHMETIC. 


formed  is  759,  and  is  read,  seven  hundred  and  fifty- 
nine.  By  placing  the  figure  0  after  the  5  in  the 
above  example,  the  same  results  are  obtained  as  re- 
gards the  figures  7  and  5.  From  the  above,  we  have 
the  following : 

For  every  period  from  right  tt)  left,  the  figure 
or  figures  increase  tenfold. 

For  every  period  from  left  to  right,  the  figure  or 
figures  decrease  tenfold. 

33.  For  convenience  in   counting,   figures   are 
divided  into  groups  of  three,  and  each  group  has  its 
particular  name  and  value. 

The  groups  are  named  as  follows : 
Hundreds,     Thousands,         Millions,          Billions, 
Trillions,        Quadrillions,      Quintillions, 
Sextillion,      Septillions,        Octillions,  etc.,  etc. 

Each  group  is  subdivided  into  units,  tens,  and 
hundreds. 

34.  To  represent  in  figures  any  number, 

1.  Place  the  figures  in  the  groups  to  which  they 
belong. 

2.  If  in  the  number  to  be  written  there  should  be 
some  vacant  spaces,  fill  those  places  with  O's.    Thus : 

Write  one  hundred  trillions,  two  hundred  and 
forty  millions,  nine  hundred  and  one  thousand,  three 
hundred  and  four. 


oj-g 
i 

Trillions. 


Ill 


Billions. 


•g-* 
&& 
2  4 

Millions. 


m 

3      4 


Thousands.        Hundreds, 
or  Units. 


NOTATION. 


19 


By  filling  the  blank  spaces  in  the  preceding 
example  with  0*s,  the  required  number  would  be 
represented  as  follows : 


1 

1 

! 

| 

1 

III 

iri 

an 

•5  .  « 

I-1! 

Crip 

SS| 

1*4 

111 

100 

000 

240 

901 

304 

Trillions. 

Billions. 

Millions. 

Thousands. 

Hundreds, 

or  Unitrf. 

and  read,  one  hundred  trillions,  no  billions,  two  hun- 
dred and  forty  millions,  nine  hundred  and  one 
thousand,  three  hundred  and  four. 

Write  in  figures : 

One  million  and  ten  thousand.       Ans.  1,010,000. 

Ten  thousand  four  hundred  and  one. 

One  hundred  trillions,  one  billion,  one  hundred 
millions,  one  thousand,  one  hundred  and  one. 

Ans.  100,001,100,001,101. 

Three  hundred  and  thirty-four  thousand,  six  hun- 
dred and  fifty-three.  Ans.  334,053. 

Three  millions,  three  hundred  and  forty-six 
thousand,  five  hundred  and  thirty-six. 

Ans.  3,346,536. 

Nine  hundred  and  three  thousand,  one  hundred 
and  forty-five.  Ans.  903,145. 

Fifty-three  billions,  four  hundred  millions,  six 
hundred  and  seventy-eight  thousand,  seven  hundred 
and  forty-two. 

Twenty-nine  millions,  three  hundred  and  sixty- 
seven  thousand  and  ninety-three. 

Twelve  thousand  and  thirteen.  Ans.  12,013. 


20  INTERMEDIATE  ARITHMETIC. 

One  hundred  and  twenty  thousand,  one  hundred 
and  thirty.  Am.  120,130. 

One  hundred  and  one  sextillions,  one  hundred 
quintillions,  two  hundred  and  forty  quadrillions, 
forty-nine  trillions,  one  billion,  twenty-one  millions, 
two  thousand  .and  ten. 

Ans.  101,100,240,049,001,021,002,010. 
Two  hundred  thousand,  one  hundred  and  one. 

Ans.  200,101. 
Ninety- eight  thousand,  seven  hundred  and  six. 

Ans.  98,706. 

Two  trillions,  twenty  billions,  two  hundred  mil- 
lions, two  thousand  and  two.  Ans. 

NOTE. — As  notation  is  so  clcsely  allied  to  numeration,  the  one 
being  the  reverse  of  the  other,  it  is  not  necessary  to  dwell  further 
on  the  subject  of  notation. 


CHAPTER    II. 
LESSON  I. 

NUMEKATION. 

35.  Numeration  is  the  art  of  reading  numbers. 

There  are  two  methods,  the  English  and  the 
French.  The  French  method  is  most  commonly 
used  in  the  United  States  and  Continental  Europe. 

§6.  As  stated  in  treating  notation,  to  form 
numbers,  figures  are  separated  into  groups  of  three 


NUMEEATION. 


21 


figures,  each  group  having  its  value,  and  each  figure, 
according  to  the  position  it  occupies  in  the  group,  is 
classed  under  the  order  or  head  of  units,  tens,  and 
hundreds. 

37.  NOTE. — To  read  numbers,  a  knowledge  of  the  art  of  numer- 
ating is  necessary,  and  to  form  numbers,  a  knowledge  of  the  art  of 
notation  is  necessary.  The  one  is  so  closely  connected  with  the 
other  that,  to  be  able  to  read  numbers,  it  is  necessary  to  be  able 
to  write  numbers,  and  vice  versa. 

38.  To  read  or  numerate,  begin  with  the  right- 
hand  figure,  or  units,  and  numerate  from  right  to 
left,  thus,  units,  tens,  hundreds,  thousands,  etc. 
Place  a  comma  after  every  third  figure  to  distin- 
guish the  group  or  set  to  which  the  figures  may 
belong  ;  then  begin  and  read  from  left  to  right,  after 
having  carefully  observed  the  order  of  the  groups. 

3O.  In  numerating,  no  notice  is  to  he  taken  of 
the  0,  further  than  to  see  that  it  occupies  its  proper 
place  among  the  figures. 


LESSON  II. 


Read  the  following  numbers  : 


1 

102030 

12 

123400 

102 

10203040 

120 

1234567 

123 

1023045067 

1023 

2034056070 

10203 

30250406070 

40503026070 

12030405060 

700800900200120 

900080007000600 

80000700006000303 

123004560007890 

2460462026404 


22 


INTERMEDIATE  ARITHMETIC. 

9876432286071 

18097642876543 

2876432154087621 

8800770066005500440033 

67008900101100121308643 

628754321865932416 

123456789123456789 

9876543210123456789 

28675432108800246189 

135792468358742651 

1010010001000010000010 


LESSON  III. 

PEOPEETIES    OF   THE  9'g. 

.  In  any  number  with  but  one  significant 
figure,  such  as  5,  50,  500,  etc.,  the  excess  of  exact  9's 
is  equal  to  the  significant  figure. 

If  we  write  any  number,  as  87643,  it  may  be  read 
in  this  way,  80  thousands,  7  thousands,  6  hundreds, 
4  tens,  or  40,  and  3  units.  Now,  the  excess  of 

9's  in  80  thousands  is 8 

in    7  thousands  is 7 

in    6  hundreds   is 6 

in    4  tens  is 4 

in    3  units  is 3 

Total,     .     .     28,  which 

is  3  nines  and  1  over,  therefore,  1  is  the  excess  of 
the  exact  number  of  9's  in  the  above  figure. 

41.  To  find  the  excess  of  9's  in  any  number, 


NUMEEATIOX.  23 

add  the  figures  comprising  the  number  together. 
Divide  the  sum  by  9.  The  remainder,  if  any,  is  the 
excess  of  9's  in  the  number.  Or — 

II.  Add  the  figures,  and  drop  9  from  the  sum  as 
soon  as  it  occurs.  Thus,  8  and  7  are  15,  9  in  15,  1 
time  and  6  over;  G  and  G  are  12,  9  in  12,  1  and  3 
over;  3  and  4  are  7  and  3  are  10,  9  in  10, 1  time  and 
1  over,  which  is  the  excess  over  the  exact  9's. 

The  same  rule  will  apply  to  all  similar  opera- 
tions. 

1.  What  is  the  excess  of  9's  in  183462  ?     Ans.  6. 

2.  What  is  the  excess  of  9's  in  10203  ?       Ans.  G. 

3.  What  is  the  excess  of  9's  in  8G401  ?        Ans.  1. 

4.  What  is  the  excess  of  9's  in  102030  ?     Ans.  G. 

5.  What  is  the  excess  of  9's  in  2345  ?         Ans.  5. 
G.  What  is  the  excess  of  9's  in  3456  ?         A?is.  0. 

7.  What  is  the  excess  of  9's  in  87643  ?       Ans.  1. 

8.  What  is  the  excess  of  9's  in  2468  ?         Ans.  2. 

9.  What  is  the  excess  of  9's  in  46810  ?        Ans.  1. 

10.  What  is  the  excess  of  9's  in  6789  ?  Ans.  3. 

11.  What  is  the  excess  of  9's  in  78910?  Ans.  7. 

12.  What  is  the  excess  of  9's  in  89123  ?  Am.  5. 

13.  What  is  the  excess  of  9's  in  91234  ?  Ans.  1. 

14.  What  is  the  excess  of  9's  in  123456  ?  Ans.  3. 

15.  What  is  the  excess  of  9's  in  234567  ?  Ans.  0. 

16.  What  is  the  excess  of  9's  in  345678  ?  Ans.  6. 

17.  What  is  the  excess  of  9's  in  456789  ?  Ans.  3. 


INTERMEDIATE  ARITHMETIC. 


CHAPTER    III. 

ADDITION. 
LESSON  I. 

42.  SIGNS. — The  sign  plus  is   the  sign  of  ad- 
dition, and  when  placed  between  two  quantities, 
signifies  that  they  are  to  be  added  together. 

43.  This  sign  (=)  is  the  sign  of  equality,  and 
signifies  that  the  quantity  or  quantities  on  each  side 
of  it  are  equal  to  each  other. 

5-f  3-f4  =  6-f4-j-2  is  read : 
5  plus  3  plus  4.  are  equal  to  6  plus  4  plus  2  ;  or, 
it  may  be  read  in  this  way,  by  substituting  the  word 
and  for  the  word  plus,  5  and  3  and  4  are  equal  to  6 
and  3  and  2. 

ADDITION  TABLE. 


2  +  1-3 

3+1=4 

4+  1=  5  1  5+  1=  6 

6+  1=  7 

7+  1=  8 

2+   2=  4 

3+   2=  5 

4+  2=  6 

5.+  2=  7 

6+  2=  8 

7+  2=  9 

2  +   3=  5 

3+  3=  6 

4+   3  =  7 

5+   3=  8 

6+  3=  9 

7+  3=10 

2+  4=  6 

3+  4=  7 

4+  4=  8 

5+  4=  9 

6+  4=10 

7+  4=11 

2  +  5=  7 

3+  5=  8 

4+  5=  9  !  5+  5=10 

6+  5=11 

7+  5=12 

2+   6=  8 

3+   6=  9 

4  +  6=10 

5+  6=11 

6+   6=12 

7+  6=13 

'2+  7=  9 

3+   7=10 

4+   7  =  11 

5+    7=12 

6+   7  =  13 

7+  7  =  14 

2  +  8  =  10 

3+  8=11 

4+  8=12 

5+   8=13 

6+  8=14 

7+  8=15 

2+  9=11 

2+10=12 

3+  9=12 
3  +  10=13 

4+   9  =  13 
4  +  10=14 

5+   9=14 
5  +  10=15 

6+  9=15 
6  +  10=16 

7+  9=16 

7  +  10=17 

2  +  11  =  13 

3  +  11=14 

4  +  11=15 

5  HI  =16 

6  +  11  =  17 

7+11=18 

2+12=14 

3  +  12=15 

4  +  12=16 

5  +  12  =  17 

6  +  12=18 

7  +  12=19 

8+  1=  9 

9+   1=10 

10+  1-11 

11+   1=12 

1=13 

8+   2=10 

9+   2=11 

10+  2=12 

11+   2=13 

2=14 

8+  3=11 

9+   3=12 

10+  3=13 

11+  3=14 

3=15 

8+  4=12    9+  4=13 

10+  4=14 

11+  4=15 

4=16 

8+  5=13 

9+   5=14 

10+  5=15 

11+   5=10 

5=17 

8+  6=14 

9+   6=15 

10+  6  =  16 

11+    6=17 

6=18 

8+  7  =  15 

9+   7=16 

10+   7  =  17 

11+   7=18 

7=19 

8+  8=16 

9+  8  =  17 

10+  8=18 

11+    8  =  19 

12+   8=20 

8+  9=17 

9+  9=18 

10+  9=19 

11+   9=20 

12+  9=21 

8  +  10=18 

9  +  10=19 

10  +  10=20 

11  +  10=21 

12  +  10=22 

8  +  11=19 

9  +  11  =  20 

10  +  11=21 

11  +  11  =  22 

12  +  11=23 

8  +  12=20 

9+12=21 

10  +  12=22 

11  +  12  =  23 

12+12=24 

ADDITION.  25 


LESSON"  II. 

44.  Addition  is  the  process  of  finding  the  sum  of 
two  or  more  numbers. 

45.  The  sum  of  two  or  more  numbers  is  a  num- 
ber which  contains  as  many  units  as  there  are  units 
in  the  several  numbers  taken  together. 

1.  What  is  the  sum  of  643  and  539  ? 

ANALYSIS. 

OPEEATION. 

Add  the  column  of  units.     If  the 
units  are  more  than  ten,  place  the  right- 
hand  figure  under  the  column  of  units,        _____ 
and  carry  the  left-hand  figure  or  figures        1182 
to  the  column  of  tens. 

Add  the  column  of  tens,  including  the  figure 
carried  from  the  units.  If  there  are  more  than  ten 
tens,  place  the  right-hand  figure  under  the  column 
of  tens,  and  carry  the  left-hand  figure  to  the  next 
column. 

Add  the  column  of  hundreds,  together  with  the 
figure  carried  from  the  tens.  If  there  are  more  than 
ten  hundreds,  place  the  right-hand  figure  under  the 
column  of  hundreds,  and  carry  the  left-hand  figure 
to  the  column  of  thousands. 

Continue  in  this  way  until  the  last  column  is 
reached  then  write  down  the  sum  of  the  last  column. 

3 


26 


INTERMEDIATE   ARITIIMETI " 


2.  Analyze  as  follows  :  write  the  )  643 

numbers — thus,                   j  539 
With  a  line  drawn  between  them, 

Add  the  column  of  the  units,  12 

Add  the  column  of  the  tens,  7 

Add  the  column  of  the  hundreds,  11 


Sum  of  the  three  columns,        1182 
3.  What  is  the  sum  of  2497,  3864,  8972  ? 


OPEKATION. 

2497 
3864 
8972 


15333 


Begin  with  the  column  of  units,  and 
add  thus:  2  and  4  are  6,  and  7  are  13 ; 
place  the  three  under  the  column,  and 
carry  the  figure  one  (mentally)  to  the 
next  column. 

Add  the  column  of  tens,  thus  :  7  and 
6  are  13,  and  9  are  22,  and  one  carried 
from  the  unit  column,  makes  23  ;  place  the  3  under 
the  column  of  tens,  and  carry  the  2. 

Add  the  column  of  hundreds:  9  and  8  are  17, 
and  4  are  21,  and  2  carried  from  the  column  of  tens 
make  23;  write  the  3  under  the  column  of  hundreds, 
and  carry  the  2  to  the  column  of  thousands. 

Add  the  column  of  thousands:  8  and  3  are  11, 
and  2  are  13,  and  2  carried  from  the  colnmn  of  hun- 
dreds make  15.  There  being  no  more  columns  to 
add,  write  the  number  15,  with  the  5  under  the 
column  of  thousands,  and  the  1  a  little  to  the  left 
of  it,  as  above. 

Hence  the  following : 


ADDITION.  27 


RULE  FOR  ADDITION. 

46.  I. — Place  the  figures  of  the  same  value  un- 
der each  other. 

II. — Begin  at  the  column  of  units,  and  add  the 
figures  found  in  that  column.  If  their  sum  exceed 
10,  place  the  right-hand  figure  under  the  column,  and 
carry  the  left-hand  one  to  the  next  column.  Con- 
tinue in  this  way  until  the  last  or  left-hand  column 
is  added.  Place  the  sum  of  the  figures  in  the  last 
column  under  the  last  column. 

The  figures  thus  found  will  be  the  answer. 


PROOF. 

47.  There  are  several  methods  of  proving  ad- 
dition. 

1.  Begin  at  the  top  and  add  the  columns  down- 
wards.    Then   begin   at  the   bottom  and   add  the 
columns  upwards.     If  the  two  results  are  the  same, 
the  addition  is  correct. 

2.  Divide  the  given  numbers  into  as  many  parts 
as  may  be  necessary.     Add  the  several  parts.     Add 
the  sums  of  the  several  parts.     If  the  sum  of  the 
several  parts  added  together  is  equal  to  the  sum 
found,  the  work  is  correct. 

3.  Find  the  excess  of  nines  in  each  of  the  num- 
bers, and  place  it  to  the  right  of  the  number.    Add  the 
numbers  thus  found,  and  note  the  excess  of  nines  in 


28 


INTERMEDIATE   ARITHMETIC. 


their  sum.  This  excess  of  nines  should  equal  the 
excess  of  nines  in  the  sum  of  the  numbers.  If  so, 
the  addition  is  correct. 


FIEST  METHOD. 

246892 
134714 
241381 
527123 


1150110 


SECOND  METHOD. 

1st  part, 

Sum  of  1st  part,       381606 

2d  part,    .     . 
Sum  of  2d  part, 


246892 
134714 


241381 
527123 

768504 


Sum  of  1st  part, 
Sum  of  2d  part, 


381606 
768504 


L  Sum  of  both  parts,  115.0110 


THIED  METHOD. 


246892 
134714 
241381 

527123 


4  excess  of  9's. 
2    "    9's. 

1  "    9's. 

2  «    9's. 


1150110 


ADDITION. 


29 


LESSON  I 

II. 

EXAMPLES. 

(1.) 

(2-) 

(3.) 

(4.) 

(5.) 

123456 

123 

12345 

689724 

18976543 

24689 

5734 

2468 

398764 

24897321 

3754 

24698 

53897 

876421 

98765428 

123 

321456 

8721 

578932 

87639742 

(6.) 

(VO 

(8.) 

(9.) 

(10.) 

5375 

25846 

93754 

13289 

52936 

1203 

64845 

49375 

91328 

65293 

3201 

52846 

54937 

89132 

36529 

3753 

85264 

75493 

28913 

93652 

5737 

42583 

37549 

32891 

29365 

(11.) 

(12.) 

(13.) 

(14.) 

(15.) 

482436 

127618 

298634 

211119 

169540 

634284 

276811 

928634 

112191 

619450 

824364 

761821 

863429 

121911 

965454 

284436 

618127 

634298 

219111 

543956 

824336 

181271 

342986 

191211 

487251 

283346 

816721 

436892 

912111 

72893 

(16.) 

(17.) 

(18.) 

(19.) 

(20.) 

246810 

246891 

1817163 

94231 

678912 

987641 

42698 

181716 

9423 

34567 

198764 

12345 

31817 

194 

89101 

387214 

2489 

1637 

32 

23456 

38761 

321 

942 

8 

78910 

30 


INTERMEDIATE  ARITHMETIC. 


LESSOR  IV. 

(21.) 
123456 

(22.) 
2345678 

(23.) 

3456789 

(24.) 

4567891 

789101 

9101234 

1012345 

8765432 

234567 

5678910 

6789101 

9101234 

891012 

1234567 

2345678 

2198765 

345678 

891012,3 

9101234 

3456789 

910123 

4567890 

5678910 

1012345 

(25.) 
1234567 

(26.) 
24689 

(27.) 
123456789 

(28.) 

3456789101 

8910123 

98642 

101234567 

2345678910 

4567891 

13579 

891012345 

1234567890 

1234567 

97513 

678910123 

2413456789 

8910123 

12345 

456789101 

1012345678 

4567890 

54321 

234567891 

9101234567 

1234567 

67890 
9876 

12345678 
91012345 

8910123456 
7891012345 
6789101234 

5678910123 

(29.) 
1234567 

(30.) 

2468972 

(31.) 
12345 

(32.) 

24764283 

8910123 

1234567 

45678 

12345678 

4567891 

8910111 

78910 

91234567 

0123456 

2345678 

10112 

89012345 

7891012 

9101112 

13456 

67890123 

3456789 

3456789 

56789 

45678901 

1012345 

6789101 

1011123 
4567890 

89108 

23456789 
12345678 
91012345 

ADDITION. 


31 


(33.) 

(34.) 

(35.) 

(36.) 

24689754 

3132333435 

246892 

36987 

32123450 

G373839404 

345678 

78963 

78910111 

1424344450 

123456 

12345 

23456178 

4748495051 

789101 

67891 

19202122 

5253545556 

121356 

23456 

2324252G 

5758590001 

789321 

78910 

27282930 

6263656667 

234567 

24689 

31323334 

87654 

32112 

34567 

89101 

(37.) 

(38.) 

(39.) 

(40.) 

2222222 

3333333 

444444 

124689 

3333333 

4445555 

444555 

44444 

4444444 

6666777 

555666 

55555 

5555555 

7888999 

666777 

66777 

C666666 

2223333 

777888 

88999 

7777777 

4444444 

888999 

11222 

8888888 

5556606 

111222 

33444 

9999999 

7778888 

222333 

55666 

1111111 

9876543 

444555 

666777 

32 


INTERMEDIATE    ARITHMETIC. 


LESSON  V. 


(41.)   (42.)   (43.) 

(44.) 

(45.)      (46.) 

867    798    321 

12386 

168872    168321 

673    765    748 

2497 

241387     54592 

571    743    617 

3874 

89764     18764 

876    435    748 

2976 

2138      2468 

901    876    286 

8397 

976      1234 

931 

2178 

22       456 

67 

8 

(47.)     (48.) 

(49.) 

(50.)     (51.) 

1       2 

3 

6    387462 

12       22 

45 

78    59321 

123      223 

678 

910     8764 

1234     2233 

9101 

1112      213 

12345    22333 

23456 

13141       89 

123456   223344 

789102 

151621 

123456 

24899 

(52.) 

(53.) 

(54) 

87649231     123456789 

87643212345 

4156789 

123456 

6789023456 

198642 

43210987 

678912345 

23456 

5678902 

67891234 

1987 

654321 

5678912 

234 

ADDITION. 


33 


(55.) 

(56.) 

(57.) 

(58.) 

(59.) 

(60.) 

12 

123 

1234 

12345 

45321 

10 

21 

456 

5678 

67891 

65432 

100 

22 

789 

9101 

02123 

78910 

1010 

23 

101 

2345 

45678 

54321 

10101 

24 

234 

6789 

90123 

67890 

100101 

25 

567 

1012 

45678 

98642 

12345 

26 

890 

3456 

91012 

12345 

12354 

27 

123 

7890 

34567 

19876 

32145 

72 

456 

1234 

89101 

187643 

13254 

82 

789 

8976 

23450 

21761 

28 

101 

2468 

32 

234 

61.  In  the  State  of  Maine  there  are  16  counties, 
in  New  Hampshire  10,  in  Vermont  14,  in  Massachu- 
setts 14,  in  Rhode  Island  5,  in  Connecticut  8.     How 
many  counties  in  the  six  New  England  States  ? 

Ans. 

62.  In  New  York  there  are  60  counties,  in  New 
Jersey  21,  in  Pennsylvania  66,  in  Delaware  3.     How 
many  counties  in  the  four  Middle  States  ? 

Ans. 

63.  New  York  City  has  a  population  of  805651 
persons,  Philadelphia  565531,  Brooklyn  266664,  Bal- 
timore 212419.     How  many  inhabitants  in  the  four 
cities  ?  Ans. 

64.  New  Orleans  has  168472  inhabitants,  Cincin- 
nati 161044,  St.  Louis  151780,  Chicago  109263,  Buf- 
falo 81131,  Newark  71941,  Louisville  69470,  Albany 


34 


INTERMEDIATE   ARITHMETIC. 


62368.     How  many  inhabitants  in  all  of  the  above 
cities?  Ans.  875469. 

65.  London  has   2803034  inhabitants,  Paris  has 
1696141,   Pekin    2000000,  Yeddo    1500000,   Canton 
1000000,  Constantinople  787000,  Berlin  547571,  York 
40377,  Londonderry  19888.     How  many  inhabitants 
in  all  of  the  above  cities?  Ans.  10394011. 

66.  Mt.  St.  Elias  is  17860  feet  high;  Mt.  Fair- 
weather,   14900  feet;  Mt.  Illaman,  12066  feet;  Mt. 
Brown,   15990  feet;  Mt.  Hooker,  15675;  Fremont's 
Peak,  13570.    If  they  were  piled  one  on  the  other, 
how  high  would  they  reach.  Ans.  90061. 

67.  The  Missouri  is  3100  miles  long;  the  Missis- 
sippi, 3160  miles;  Mackenzie,  2440  miles;  St.  Law- 
rence, 2200  miles ;  Arkansas,  2170  miles ;  Rio  Grande, 
1800  miles;  Red,   1500;    Nebraska,   1500;    Nelson, 
1600;  Columbia,  1200;  Colorado,  1000;  Ohio,  948; 
Lewis,  900 ;  Tennessee,  800.     If  they  were   placed 
one  after  the  other,  forming  one  long  river,  how  many 
miles  in  length  would  it  be  ?  Ans.  24318. 

68.  The  Amazon  is  4000  miles ;   La  Plata,  2250 
miles;  Madeira,  1800  miles;  Paraguay,  1600  miles; 
Orinoco,  1590;  St.  Francisco,  1260;  Tocantins,  1100. 
How  many  miles  in  length  would  they  be  if  formed 
into  one  stream  ?  Ans.  13600. 

69.  Asia    has    696725000    inhabitants;    Africa, 
70670000;  North  America,  49280480 ;  South  Amer- 
ica, 21356870  ;  Oceanica,  25898788;  Europe,  282959- 
505.     How  many  inhabitants  on  the  globe  ? 

Ans.  1146890643. 

70.  Russia  has   66891493   inhabitants ;    Sweden 


ADDITION.  35 

and  Norway,  5474452 ;  Austria,  35019058;  France, 
37472132;  Turkey,  15500000;  Spain,  16560813; 
Great  Britain  and  Ireland,  29335340 ;  Italy,  22430000 ; 
Prussia,  23810743;  Germany,  13774510;  Portugal, 
3923410;  Denmark,  1592354;  Greece,  1343293; 
Switzerland,  2510394;  Holland,  3569456;  Belgium, 
4731957.  How  many  inhabitants  in  Europe  ? 

Ans.  282959505. 

71.  China  has  a  population  of*  408000000  inhab- 
itants; Siberia,  4600000 ;  Hindoostan,  183500000; 
Arabia,  8000000 ;  Farther  India,  20000000 ;  Inde- 
pendent Tartary,  6000000 ;  Turkey  in  Asia,  15000000 ; 
Persia,  9000000  ;  Japan  Empire,  30000000  ;  Afghan- 
istan,  6000000;  Beloochistan,  2000000;  Georgia, 
1625000;  Corea,  2000000;  Ladak,  1000000.  How 
many  inhabitants  in  Asia  ?  Ans.  696725000. 

48.  To  add  from  left  to  right  and  from  right  to 
left  alternately. 


1.  Add  241,  362,  648,  and  182. 
1st.    Arrange   the    numbers    one 


under  the  other,  as  in  the  operation; 


begin  with  the  first  number  and  read 
as  follows :  241  and  2  units  make  243 
units,  and  60  units  make  303  units, 
and  300  units  make  603  units.  603 


OPERATION. 


362 

648 

182 

1433 


units  and  8  units  make  611  units,  and 
40  units  make  651  units,  and  600  units  make  1251 
units.  1251  units  and  2  units  make  1253  units,  and 
80  units  make  1333  units,  and  100  units  make  1433 
units,  the  answer. 


36 


INTERMEDIATE  ARITHMETIC. 


Leaving  the  word  units  out,  it  is  read  more  rap- 
idly in  this  way. 

241  and  2  are  243,  243  and  60  are  303,  303  and 
300  are  603,  603  and  8  are  611,  611  and  40  are  651, 
651  and  600  are  1251,  1251  and  2  are  1253,  1253  and 
80  are  1333,  1333  and  100  are  1433,  the  answer. 

49.  Whole  columns  of  a  ledger  can  be  added  in 
this  way  very  rapidly  and  correctly. 

*5O.  The  examples  given  are  simple  enough  to  be 
performed  by  almost  any  beginner. 

EXAMPLES. 

(1.)  (2.)  (3.)  (4.)  (5.)  (6.)  (7.)  (8.)  (9.) 

11  11  12  14  15  16  17  18  19 

12  12  13  15  16  17  18  19  20 

13  31  14  16  17  18  19  20  21 

14  41  15  17  18  19  20  21  22 

(10.)  (11.)  (12.)  (13.)  (14.)  (15.)  (16.)  (17.)  (18.) 


20 

21 

23 

24 

25 

26 

27 

28 

29 

21 

22 

24 

25 

26 

27 

28 

29 

30 

22 

23 

25 

26 

27 

28 

29 

30 

31 

23 

24 

26 

27 

28 

29 

30 

31 

32 

24 

25 

27 

28 

29 

30 

31 

32 

33 

(19.)  (20.)  (21.)  (22.)  (23.)  (24.)  (25.)  (26.) 

Ill  222  333  444  555  666  777  888 

222  333  444  555  666  777  888  999 

333  444  555  666  777  888  999  111 

444  555  666  777  888  999  111  222 

555  666  777  888  999  111  222  333 


SUBTRACTION. 


37 


(27.) 

(28.) 

(29.) 

(30.) 

9999 

11111 

222222 

123456 

1111 

22222 

333333 

78912 

2222 

33333 

444444 

12345 

3333 

44444 

555555 

6789 

4444 

55555 

66666G 

123 

(31.) 

(32.) 

(33.) 

(34.) 

12345 

54321 

987643 

123456 

67891 

19876 

543212 

789012 

23045 

54032 

678901 

345678] 

67890 

19876 

765432 

901234 

12345 

54321 

890123 

908765 

67890 

67890 

654321 

123456 

12345 

12345 

789012 

789012 

67890 

69876 

345678 

123465 

CHAPTER  IV. 

SUBTRACTION. 
LESSON  I. 

51.  SUBTRACTION  is  the  process  of  finding  the 
difference  between  two  numbers.     Thus,  5  from  7 
leaves  2. 

52.  In  the  above  example,  the  figure  5  is  the 

4 


38 


INTERMEDIATE  ARITHMETIC. 


subtrahend,  the  figure  7  the  minuend,  and  the  figure 
2  the  remainder. 

53.  The  sign  (—  )  minus  is  the  sign  of  subtrac- 
tion, and  when  it  is  placed  between  two  quantities 
signifies  that  they  are  to  be  subtracted. 

54.  The  minuend  is  placed  on  the  left  and  the 
subtrahend  on  the  right  of  the  sign. 

55.  When  separated  by  the  sign  it  is  read  thus  : 


7  minus  5  equals  2  ;  or,  better,  5  from  1  leaves  2. 

56.  The  pupil  should  commit  to  memory  the 
following 

TABLE   OF   SUBTKACTIOST. 


2-1=  1 

2—2=  0 

3—3=  0 

4-4=  0 

5-5=  0 

6-6=  0 

3-1=  2 

3-2=  1 

4-3=  1 

5-4=  1 

6—5=  1 

7-6=  1 

4-1=  3 

4—2=  2 

5—3=  2 

6-4=  2 

7-5=  2 

8—6=  2 

5-1=  4 

5-2=  3 

6-3=  3 

7-4=  3 

8-5=  3 

9—6=  3 

6-1=  5 

6-2=  4 

7-3=  4 

8-4=  4 

9—5=  4 

10-6=  4 

7-1=  6 

7-2=  5 

8-3=  5 

9-4=  5 

10—5=  5 

11—6=  5 

8-1=  7 

8-2=  6 

9-3=  6 

10-4=  6 

11—5=  6 

12—6=  6 

9-1=  8 

9-2=  7 

10-3=  7 

11-4=  7 

12-5=  7 

13-6=  7 

10-1=  9 

10-2=  8 

11-3=  8 

12-4=  8 

13-5=  8 

14—6=  8 

11-1=10 

11-2=  9 

12-3=  9 

13—4=  9 

14-5=  9 

15-6=  9 

12-1=11 

12-2=10 

13-3=10 

14-4=10 

15—5=10 

16-6=10 

i 

7-7=  0 

8-8=  0 

9-9-  0 

10-10=  0 

11—11=  0 

12—12=  0 

8-7=  1 

9-8=  1 

10-9=  1 

11-10=  1 

12—11=  1 

13—12=  1 

9—7=  2 

10-8=  2 

11-9=  2 

12—10=  2 

13—11=  2 

14—12=  2 

10-7=  3 

11-8=  3 

12-9=  3 

13—10=  3 

14—11=  3 

15—12=  3 

11-7=  4 

12—8=  4 

13-9=  4 

14—10=  4 

15—11=  4 

16-  -12=  4 

12-7=  5 

13—8=  5 

14-9=  5 

15—10=  5 

16-11=  5 

17—12=  5 

13-7=  6 

14-8=  6 

15-9=  6 

16—10=  6 

17—11=  6 

18—12=  6 

14-7=  7 

15-8=  7 

10-9=  7 

17—10=  7 

18—11=  7 

19—12=  7 

15—7=  8 

16-8=  8 

17—9=  8 

18-10=  8 

19—11=  8 

20—12=  8 

16-7=  9 

17—8=  9 

18—9=  9 

19-10=  9 

20—11=  9 

21—12=  9 

17-7=10 

18-8=10 

19-9=10 

20-10=10 

21—11=10 

22—12=10 

57.  To  subtract  one  number  from  another,  both 
of  them  must  be  of  the  same  kind. 


SUBTRACTION.  39 

Thus,  1  inches  could  not  bo  subtracted  from  7 
feet ;  but  we  could  reduce  7  feet  in  the  following 
manner,  0  feet  12  inches,  which  would  be  of  the 
same  value  as  7  feet.  Now  subtract  7  inches  from 
the  12  inches,  and  there  remain  G  feet  5  inches. 

The  proof  of  subtraction  depends  upon  this  prin- 
ciple : 

58.  That  the  greater  number  is  equal  to  the 
smaller  number  and  the  difference  added  together. 

LESSON  IT. 

50.  When  but  one  figure  is  used  in  both  min- 
uend and  subtrahend : 

Place  the  greater  number  above  the  less,  with  a 
line  drawn  below  them.  Under  this  line  place  the 
remainder  or  difference. 

MENTAL  EXEKCISES. 

7  from  8  leaves  how  many  ?  5  from  G  ?  3  from  7  ? 

9  from  10  ?  8  from  10  ?  2  from  G  ?  5  from  8  ? 
3  from  7  ? 

2  from  G  ?  1  from  3  ?  5  from  7  ?  9  from  10  ? 
7  from  9  ? 

1  from  2  ?     2  from  4  ?     3  from  6  ?     4  from  8  ? 
5  from  10? 

2  from   3  ?     2   from  5  ?     2   from  7  ?     2  from  9  ? 

2  from  8  ? 

3  from  4  ?     3  from  5  ?     3  from  G  ?     3  from  7  ? 

3  from  8  ? 

4  from  5  ?     4  from  6  ?     4  from  7  ?    4  from  9  ? 

4  from  8  ? 


40 


INTERMEDIATE  ARITHMETIC. 


LESSON  III. 

60.  When  several  figures  are  employed  in  both 
minuend  and  subtrahend. 

CASE  I. 

When  the  figures  of  the  subtrahend  are  of  less 
value  than  the  corresponding  ones  in  the  minuend. 

RULE. 

Subtract  the  figures  in  the  subtrahend  from  thosz 
in  the  minuend)  and  place  the  difference  below  the  line 
drawn  under  the  subtrahend. 

61.  The  pupil  must  be  careful  and  see  that  his 
units  tens,  hundreds,  etc.,  are  in  their  proper  places. 


(1.) 

456 
123 


(2.) 

987 
432 


EXERCISES. 

(3.)             (4.)  (5.) 

79318         8987  5432 

58107         2461  1321 


(6.) 

8976898 
7265787 


(7.)    (8.)     (9.)     (10.)    (11.)     (12.) 

8976   64389   876492   88888   99999   77778888 
7425   52273   765381   24681   36901   45632547 


(13.)   (14.)    (15.)     (16.)    (17.)     (18.) 
8889  879642   976384    7643   87649   28976438 
7786  768221   251271    6221   72116   12345216 


SUBTRACTION. 


41 


(19.)   (20.)    (21.)     (22.)   (23.)     (24.) 


98764  876213   76439 
72413  245102   51216 


8749   87924   98769876 
2513   12312   12345672 


(25.)    (26.)     (27.)    (28.)   (29.)     (30.) 

98764  5987689  897643  89769   76834   5986873 
24673  1276543  245121  21212   12121   2211331 


(31.)   (32.)    (33.)    (34.)    (35.)     (36.) 

8976  897643  123456   34567   88992   98765482 
1312  253121   12345    3456   24681   12345121 


(40.)  (41.)      (42.) 

8976  98764  8976543 

3452  72315  5312312 

(46.)  (47.)  (48.) 

98764  87649  9876543 

21213  25517  2123431 


(52.)    (53.)     (54) 

87643   99889   88888888 
24612   24683   12345678 


(37.) 

(38.) 

(39.) 

9876 

8765 

56789 

5432 

4321 

23456 

(43.) 

(44.) 

(45.) 

8976 

89764 

98765 

5213 

25761 

43212 

(49.)' 

(50.) 

(51.) 

9876 

87643 

6743218 

7253 

35213 

2521116 

42 


IISTTEKMEDIATE    ARITHMETIC. 


(55.)   (56.) 

2468   4689 
1234   3456 


(5V.) 

678912 
123411 


(58.) 
24891 
12341 


(61.)     (62.) 

9898    87878 
8888    77777 


(63.)  ' 

67676767 
66666666 

(66.) 

77777777777 
12345671234 


(59.) 
89764 
72213 

(60.) 
99999999 
23456789 

(64.) 
89764 
32512 

(65.) 

289764 
123452 

LESSON  IV. 

CASE   II. 


63.  When  the  figures  in  the  subtrahend  are  in 
some  instances  greater  than  the  corresponding  ones 
in  the  minuend,  proceed  as  follows  : 

Write  the  less  number  under  the  greater,  with  a 
line  drawn  beneath  the  lower  one,  as  in  the  following 


EXAMPLE. 

Minuend.          3687421 
Subtrahend.     2758622 


Remainder. 


3    6973 


928799 


Begin  and  subtract:  2  from  1,  I  cannot;  borrow 
1  ten,  which  is  equal  to  10  units,  and  1  unit  in  the 
minuend  makes  11.  2  from  11  leaves  9.  Place  the 


SUBTRACTION.  43 

9  under  the  column  of  units,  and  add  1  to  the  tens 
of  the  subtrahend  (or  subtract  1  from  the  tens  of  the 
minuend),  which  would  make  3  tens  instead  of  2  tens. 
To  subtract  the  3  from  the  2  would  be  impossible; 
but  borrow  1  from  the  column  of  hundreds,  which  is 
equal  to  10  tens,  and  2  tens  gives  12  tens,  from  which 
subtract  the  3,  leaving  9.  Write  the  figure  9  under 
the  column  of  tens,  and  add  1  to  the  figure  in  the 
column  of  hundreds,  which  would  give  7  hundreds. 
To  subtract  the  7  hundreds  from  the  4  hundreds 
would  be  impossible;  but  borrow  1  from  the  col- 
umn of  thousands,  which  would  be  equal  to  ten 
hundreds,  and  4  hundreds  gives  14  hundreds ;  sub- 
tract 7  from  14,  and  7  remains.  Write  the  figure  7 
under  the  column  of  hundreds,  and  add  1  to  the 
figure  in  the  column  of  thousands,  making  it  9. 
Subtract  the  column  of  thousands.  9  from  7  impos- 
sible; borrow  1  ten  thousand,  which  is  equal  to  10 
one  thousands,  and  7  gives  17  thousands.  9  from  17 
leaves  8.  Write  the  figure  8  under  the  column  of 
thousands,  and  add  1  to  the  next  ligure  of  the  sub- 
trahend, making  it  6.  Subtract :  6  from  8  leaves  2  ; 
write  the  figure  2  under  the  column  of  tens  of  thous- 
ands. In  this  case  we  borrowed  nothing  from  the 
next  column ;  therefore  we  have  nothing  to  add  to 
the  column  of  hundreds  of  thousands,  but  merely  to 
subtract  the  figures  as  we  find  them.  Subtract:  7 
from  6,  impossible ;  borrow  1  from  the  next  column 
of  millions,  which  is  equal  to  10  hundreds  of  thou- 
sands, and  6  making  16  hundreds  of  thousands.  7 
from  16  leaves  9.  Write  the  figure  9  under  the  col- 


44  INTERMEDIATE  ARITHMETIC. 

umn  of  hundreds  of  thousands,  and  add  1  to  the  next 
figure  in  the  subtrahend,  making  it  3.  Subtract:  3 
from  3  leaves  0,  which  figure  is  never  written  to  the 
left  of  any  integral  number.  The  answer  is  928799. 
To  prove  subtraction,  add  the  remainder  to  the 
subtrahend,  and  the  result  must  be  equal  to  the 
minuend. 

Subtrahend.     2758622 
Remainder.        928799 

Minuend.          3687421 

The  pupil  should  be  thoroughly  drilled  in  reading 
subtraction.  Thus : 

From  786433  take  536248. 

EXAMPLE. 

786433 
596248 

190185 

BEADING. 

8  from  13  leaves  5,  write  down  the  5  and  carry  1. 
1  and  4  are  5.  5  from  13  leaves  8,  write  the  8,  and 
carry  1.  1  and  2  are  3.  3  from  4  leaves  1.  (Since 
we  borrowed  nothing,  we  have  nothing  to  carry.) 
6  from  6  leaves  0,  write  down  the  0.  9  from  18 
leaves  9,  write  the  figure  9  and  carry  1.  1  and  5  are 
6.  6  from  7  leaves  1.  Write  down  the  1. 


SUBTRACTION.  45 

PROOF. 

The  answer  would  be     190185     Remainder. 
596248     Subtrahend. 

'786433     Minuend. 
From  which  we  have  the  following  general 

RULE. 

63. — I.  Write  the  greater  number  above  the  less, 
with  a  line  drawn  beneath  them. 

II.  Begin  with  the  units,  and  subtract.    If  the 
figure  in  the  subtrahend  is  larger  than  the  figure  in 
the  minuend,  add  10  to  it  (mentally},  and  subtract. 

III.  Add  1  to  the  next  figure  in  the  subtrahend, 
and  subtract.    If  the  figure  in  the  subtrahend  is  again 
too  large,  add  10  to  it  mentally,  and  subtract. 

IV.  Proceed  in  this  way  until  all  of  the  figures  in 
the  subtrahend  are  subtracted  from  the  corresponding 
figures  in  the  'minuend. 

64.  XOTE.— When  it  becomes  necessary  to  add  10  mentally  to 
the  figure  in  the  minuend,  1  must  be  added  to  the  next  figure  of 
the  subtrahend,  but  in  no  other  case. 


EXAMPLES. 

(2.)                 (3.)  (4.)  (5.) 

123456    897643  888888  2999919 

122567    188754  99999  898721 


46 


INTERMEDIATE   ARITHMETIC. 


(6.)       (7.) 

89778842   246891 
8898953    157802 


(8.)       (9.).      (10.) 

] 23456    345678    456789 
34567     56789     67891 


(11.) 

(12.) 

(13.) 

(14.) 

(15.) 

1234567 

123456789 

223344 

334455 

445566 

345678 

34567891 

33445 

44556 

55667 

(16.)     (17.) 

556677   667788 
66778    77889 


(18.) 

778899332 
88993323 


(19.)      (20.) 

889911   991122 
99122    99112 


(21.)       (22.)       (23.)       (24.) 

1122334    1223345    2233456    13579246 
223345     334455     334457     5689357 


(25.)     (26.) 


(27.) 


(28.) 


(29.) 


246892   123456   3456789   8764312   268972^ 
57903     13578    567890    876431     788935 


(30.)         (31.)       (32.)      (33.) 

8899776655    7642839    2286931    1869741 
998877666     764289     121286     979852 


SUBTRACTION. 


47 


(35.) 

228769 
39892 


(39.)      (40.) 

200020    202020 
20002     20202 


(36.) 

18976542 
9897653 

(41.) 

3030303 
303030 


(42.) 

40404040404040 
4040404040404 


(43.) 

50505050 
6070809 


(44.)       (45.)       (46.) 

6060606    70809010    6243875 
708098     7080908     624897 


(47.) 

(48.) 

(49.) 

(50.) 

2030020 

30405060 

70809    1000000 

2000000 

203103 

3405106 

70080 

246801 

234567 

(51.) 

(52.) 

(53.) 

(54.) 

30000000 

40000000 

1234567 

10203040506070 

11234567 

12357982 

891873 

1234567897082 

LESSON  V. 

55.  Asia  has  696725000  inhabitants.     Europe  has 
282959505.     How  many  more  has  Asia  than  Europe  ? 

Ans.  413765495. 

56.  North   America    has   49280480   inhabitants. 


48 


INTERMEDIATE    ARITHMETIC. 


South   America  has    21356870   inhabitants.      How 
many  more  has  North  America  than  South  America  ? 

Ans.  27923610 

57.  The  area  of  the  United  States  is   3002332 
square    miles.      British    America,    2914318     square 
miles.     How  much  larger  is  the  United  States  than 
British  America?  Ans.  88014  square  miles. 

58.  The  area  of  Central  America  is  219000  square 
miles.     The  West  Indies,  92820  square  miles.     How 
much  larger  is   Central  America  than    the  West 
Indies?  Ans.  126180  square  miles. 

59.  Massachusetts     has     1231065     inhabitants. 
Louisiana,  708002.     How  many  more  has  Massachu- 
setts than  Louisiana  ?  Ans.  523063. 

60.  The  area  of  Louisiana  is  46431  square  miles. 
Massachusetts,  7800  square  miles.     How  much  larger 
is  Louisiana  than  Massachusetts  ? 

Ans.  38631  square  miles. 

61.  London  has  two  millions  eight  hundred  and 
three  thousand  and  thirty-four  inhabitants,  and  Paris 
has  one  million  six  hundred  and  ninety-six  thousand 
one  hundred  and  forty-one  inhabitants.     How  many 
more  has  London  than  Paris?  Ans.  1106893. 

62.  Constantinople  has  seven  hundred  and  eighty- 
seven  thousand  inhabitants.     New  York  has  eight 
hundred  and  five  thousand  six  hundred  and  fifty-one 
inhabitants.     Has  New  York  more  than  Constanti- 
nople? Ans.  18651  more. 

63.  From  one  billion   take  two  hundred  and 
ninety-nine  millions  nine  hundred  and  ninety-nine. 

Am.  700999001. 


MULTIPLICATION.  49 

CHAPTER  IV. 

MULTIPLICATION. 
LESSON  I. 

65.  MULTIPLICATION  is  the  process  of  taking  one 
number  as  many  times  as  there  are  units  in  another. 

66.  When   a  number  is  to  be  added  to  itself 
several  times,  the  operation  is  shortened  by  the  pro- 
cess of  multiplication. 

67.  In  multiplication,  three  terms  are  used :  the 
multiplicand,  the  multiplier,  and  the  product. 

68.  The  multiplicand  is  the  number  to  be  multi- 
plied. 

6O.  The  multiplier  is  the  number  by  which  we 
multiply  the  multiplicand. 

70.  The  product  is  the  result  of  the  multiplica- 
tion. 

71.  The  multiplicand  may  be  either  an  abstract 
or  a  concrete  number.     The  multiplier  must  always 
be  considered  as  an  abstract  number. 

72.  The  product  is  always  of  the  same  kind  as 
the  multiplicand. 

EXAMPLE. 

$40  X  3  =  $120 

That  is,  3  times  40  dollars  are  120  dollars. 

In  this  example,  40  dollars  is  the  multiplicand,  3 
the  multiplier,  and  120  dollars  the  product. 


50 


INTERMEDIATE    ARITHMETIC. 


73.  The  sign  of  multiplication  is  formed  by  two 
small  lines  crossing  each  other  so  as  to  form  an  X  • 

74.  It  shows  that  the  two   numbers   between 
which  it  is  placed  are  to  be  multiplied  together. 

Thus,  6XV=42  is  read,  6  times  7  are  42. 


EXAMPLE. 

If  1  barrel  of  flour  cost  8  dollars,  how  much  will 
5  barrels  cost  ? 

It  is  plain  that  if  one  barrel  would  cost  eight 
dollars,  5  barrels  would  cost  5  times  as  much  as  1 
barrel,  or  5  X  8=40  dollars.  Or, 

If  1  barrel  cost  8  dollars,  5  barrels  would  cost  5 
times  8,  or  8+8+8  +  8  +  8,  which  added  together 
gives  40. 

TABLE   OF  MULTIPLICATION. 


2x   1=  2 

3x   1=  3 

4x   1=  4 

5x   1=  5 

6x   1=  6 

7x   1=  7 

2x   2=  4 

3x   2=  6 

4x   2=  8 

5x   2=10 

6x   2=12 

7x   2=14 

2x   3=:  6 

3x   3=  9 

4x   3=12 

5x   3=15 

6x   3=18 

7x   3=21 

2x   4=  8 

3x   4=12 

4x   4=16 

5x   4=20 

6x   4=24 

7x  4=28 

2x   5=10 

3x   5=15 

4x   5=20 

5x   5=25 

6x   5=30 

7x   5=35 

2x   6=12 

3x   6=18 

4x   6=24 

5x   6=30 

6x   6=36 

7x   6=42 

2x   7  =  14 

3x   7=21 

4x   7=28 

5x   7=35 

6x   7=42 

7x   7=49 

2x   8=16 

3x   8=24 

4x    8=32 

ox   8=40 

6x   8=48 

7x   8=56 

2x   9=18 

3x   9=27 

4x    9=36 

5x   9=45 

6x    9=54 

7x    9=63 

2x10=20 

3x10=30 

4  x  10=40 

5  x  10=50 

6  x  10=60 

7  x  10=70 

2x11=22 

3x11=33 

4  x  11=44 

5x11  =  55 

6  x  11=66 

7x11=77 

2x12  =  24 

3x12=36 

4x12=48 

5  x  12=60 

6x12=72 

7x12=84 

8x   1=  8 

9x   1-     9 

10  x   1=  10 

11  x   1=  11 

12  x   1=  12 

8x   2=16 

9x  2=  18 

10  x   2=  20 

11  x    2=  22 

12x   2=  24 

8x   3=24 

9x   3=  27 

10  x   3=  30 

11  x   3=  33 

12  x   3=  36 

8x   4=32 

9  x   4=  36 

10  x  4=  40 

11  x   4=  44 

12  x  4=  48 

8x   5=40 

9x   5=  45 

10  x   5=  50 

11  x  5=  55 

12  x   5=  60 

8x   6=48 

9x    6=  54 

10  x   6=  60 

11  x   6=  66 

12  x   6=  72 

8x    7=56 

9x   7=  63 

10  x    7=  70 

llx   7=  77 

12  x   7=  84 

8x   8  =  64 

9x   8=  72 

10  x    8=  80 

11  x   8=  88 

12  x   8=  96 

8x   9=72 

9x   9=  81 

10  x    9=  90 

llx   9=  99 

12  x   9=108 

8x10=80 

9x10=  90 

10  x  10=100 

11x10=110 

12x10=120 

8x11=88 

9x11=  99 

10x11=110 

11x11=121 

12x11  =  132 

8x12=96 

9x12=108 

10x12=120 

11x12=132 

12x12=144 

MULTIPLICATION.  51 


LESSON  II. 

CASE   I. 

75.  When  the  multiplicand  and  multiplier  arc 
between  1  and  9  inclusive. 

I 

EULE. 
Multiply  as  in  the  table. 

Thus: 

1.  What  cost  5  hats  @  $3  ? 

Ans.  5  x  3  =  15  dollars. 

2.  What   cost   6   hats   @  $2  ?     @  $3  ?     @  $4  ? 

@  $5  ?     @  $6  ?     @  $7  ?     @  $8  ? 

3.  What  cost  2  slates  @  4  cents  ?     @  5  cents  ? 
@G  cents?     @  7  cents?     @  8  cents?     @  9  cents? 

4.  if  1  pound  of  sugar  cost  8  cents,  what  will  2 
pounds  cost  ?     3  pounds  ?     4  pounds  ?     5  pounds  ? 
6  pounds  ?     7  pounds  ?     8  pounds  ? 

5.  A  good  pair  of  shoes  is  worth  8  dollars,  what 
must  I  give  for  2  pairs  ?   3  pairs  ?   4  pairs  ?  5  pairs  ? 
6  pairs  ?     7  pairs  ?     8  pairs  ?     9  pairs  ? 

6.  If  a  man  earns  6  dollars  in  1  week,  how  much 
will  he  earn  in  2  weeks  ?     3  weeks  ?     4  weeks  ?     5 
weeks  ?    6  weeks  ?     7"  weeks  ?    8  weeks  ?    9  weeks  ? 

7.  If  1  thousand  feet  of  lumber  cost  12  dollars, 
what  cost  2  thousand  ?     3  thousand  ?     4  thousand  ? 


52 


INTERMEDIATE    ARITHMETIC. 


5  thousand  ?  G  thousand  ?  7  thousand  ?  8  thousand  ? 
9  thousand? 

8.  If  1  bottle  of  ink  cost  5  cents,  what  cost  2  bot- 
tles ?    3  bottles  ?    4  bottles  ?    5  bottles  ?    6  bottles  ? 
7  bottles  ?     8  bottles  ?     9  bottles  ? 

9.  If  1  bushel  of  apples  cost  2  dollars,  what  cost 
2   bushels?     3   bushels?     4  bushels?     5   bushels? 

6  bushels  ?     7  bushels  ?     8  bushels  ?     9  bushels  ? 

10.  If  1  hat  is  worth  8  dollars,  what  are  9  hats 
worth  ? 

11.  If  1  box  of  cheese  cost  9  dollars,  what  cost  9 
boxes  ? 

12.  If  a  man  earns  7  dollars  in  1  week,  how  much 
will  he  earn  in  7  weeks  ? 

13.  If  1  barrel  of  potatoes  cost  5  dollars,  what 
will  8  barrels  cost  ? 

14.  If  a  car  travels  7  miles  an  hour,  how  far  will 
it  travel  in  4  hours  ? 

15.  If  1  pair  of  pantaloons  cost  6  dollars,  what 
will  6  pair  cost  ? 

16.  If  1  yard  of  cotton  cost  9  cents,  what  will  8 
yards  cost  ? 

LESSON  III. 

CASE   II. 

TO.  When  there  are  several  figures  in  the  multi- 
plicand and  only  one  in  the  multiplier. 


MULTIPLICATION.  53 


EXAMPLE. 

Multiply  2348  by  5. 

7T.    Multiply    each    and    every 


OPERATION. 

number  in  the   multiplicand  by  tho 

multiplier.     Begin  thus :    5  times  8 

are  40,  or  4  tens  and  0  units.     Write 

the  figure  0  under  the  figure  of  units  11740 

and  carry  the  4  tens.     5  times  4  are 

20. and  4  are  24,  or  2  hundreds  and  4  tens.     Write 

the  figure  4  under  the  tens  of  the  multiplicand  and 

carry  the   2.     5   times  3   are   15   and  2  are  17,  or  1 

thousand  and  7  hundreds.     Write  the  figure  7  under 

the  column  of  hundreds  and  carry  the  figure  1.     5 

times  2  are  10  and  1  are  11.     There  being  no  more 

figures  in  the  multiplicand,  write  down  the  number 

found  by  the  last  multiplication. 

EXAMPLES. 

1.  Multiply  12345  by  3.  Ans.  37035. 

2.  Multiply  23456  by  4.  Ans.  93824. 

3.  Multiply  34567  by  5.  Ans.  172835. 

4.  Multiply  45678  by  6.  Ans.  274068. 

5.  Multiply  56789  by  7.  Ans.  397523. 

6.  Multiply  67891  by  8.  Ans.  543128. 

7.  Multiply  78912  by  9.  Ans.  710208. 

8.  Multiply  89123  by  3.  Ans.  267869. 

9.  Multiply  91234  by  4.        —•  Ans.  364936. 
10.  Multiply  1234567  by  2.  Ans.  2469134. 

5* 


54 


INTERMEDIATE    ARITHMETIC. 


Ans.  246913578. 
Ans.  370370367. 
Ans.  493827156. 
Am.  617733990. 
Ans.  5925925926. 


11.  Multiply  123456789  by  2. 

12.  Multiply  132546789  by  3. 

13.  Multiply  312546798  by  4. 

14.  Multiply  123546798  by  5. 

15.  Multiply  987654321  by  6. 

16.  Multiply  897645123  by  7.     Ans.  6283515861. 

17.  Multiply  987612345  by  8.     Ans.  7900898760. 

18.  Multiply  123456987  by  9. 

19.  Multiply  123456789  by  9. 

20.  Multiply  123456789  by  8. 

21.  Multiply  2463516789  by  7.  Ans.  17244617523. 

22.  Multiply  123456798  by  6.       Ans.  740740788. 

23.  Multiply  213546789  by  7.      Ana.  1494827523. 

24.  Multiply  312645879  by  5.     Ans.  1563229395- 

25.  Multiply  897543261  by  4.     Ana.  3590173044. 

26.  Multiply  246891357  by  3. 

27.  Multiply  462893157  by  2. 

28.  Multiply  1234579123  by  3.  Ans.  3703737369. 

29.  Multiply  123456789123  by  4. 

Ana.  493827156492. 

30.  Multiply  312654789312  by  5. 

Ana.  1563273946560. 

31.  Multiply  3124658971321  by  6. 

Ana.  18747953827926. 

32.  Multiply  18765432897654  by  7. 

Ans.  131358030283578. 


Ans.  1111112883. 

Ans.  1111111101. 

Ana.  987654312. 


Ans.  740674071. 
Ans.  925786314 


MULTIPLICATION.  55 

33.  Multiply  876154389762541  by  8. 

Ans.  7009235118100328. 

34.  Multiply  7685143987614521  by  9. 

Ans.  6910C295888530G89. 

35.  Multiply  263748591221314151  by  2. 

Ans.  527497182442628302. 

36.  Multiply  152535455565758595  by  3. 

Ans. 

37.  Multiply  223344556677889911  by  4. 

Ans. 

38.  Multiply  123321213312132231  by  5. 

Ans. 

39.  Multiply  3456435645634435654  by  6. 

Ans. 

40.  Multiply  8767789888946398498  by  7. 

Ans. 

41.  Multiply  98293849899868977998  by  8. 

Ans. 

42.  Multiply  2864988767478394987964  by  9. 

Ans. 

43.  Multiply  989898989897969587898  by  8. 

Ans. 


LESSON  IV. 

CASE  III. 

78.  When  there  are  several  figures  in  both  mul- 
tiplier and  multiplicand. 


56 


INTERMEDIATE   ARITHMETIC. 


OPEEATIOX. 


493824 
370368 
246912 

28888704 


EXAMPLE. 

Multiply  123456  by  234. 

1.  Multiply  all  the  figures  of  the 
multiplicand  by  the  figure  represent- 
ing the  units  in  the  multiplier.    Thus, 
1st,  4  times  6  are  24 ;  write  the  figure 

4  Tinder   the   column   of  units,  and 
carry  the    2.     2d,  4  times   5  are   20 
and  2  are  22 ;  write  the  figure  2  and 
carry  2.     3d,  4  times  4  are  16  and  2 
are  18;  write  the  figure  8  and  carry 
1.     4th,  4  times  3  are  12  and  1  are 

13;  write  the  figure  3  and  carry  1.  5th,  4  times  2 
are  8  and  1  are  9 ;  write  the  figure  9.  In  this  last 
multiplication  we  have  no  figure  representing  the 
next  higher  order,  therefore  we  have  nothing  to 
carry.  6th,  4  times  1  are  4^  write  the  4  a  little  to 
the  left  of  the  9. 

2.  Begin  with  the  figure  representing  the  tens, 
which  in  this  case  is  3.     Tens  by  units  give  tens. 
1st,  3  times  6  are  18 ;  write  the  8  under  the  tens,  and 
carry  1.     Tens  by  tens  give  hundreds.     2d,  3  times 

5  are  15  and  1  are  16;  write  the  figure  6  under  the 
column  of  hundreds  and  carry  1.     Tens  by  hundreds 
give  thousands.     3d,  3  times  4  are  12  and  1  are  13 ; 
write  the  figure  3  under  the  column  of  thousands  and 
carry  1.     Tens  by  thousands  give  tens  of  thousands. 
4th,  3  times  3  are  9  and  1  are  1 0 ;  write  the  figure  0 
under  the  column  of  tens  of  thousands  and  carry  1. 
Tens  by  tens  of  thousands  give  hundreds  of  thou- 


MULTIPLICATION.  57 

sands.  5th,  3  times  2  are  6  and  1  are  7.  In  this 
case  we  have  nothing  to  carry.  Tens  by  hundreds 
of  thousands  give  ten  hundreds  of  thousands,  or  mil- 
lions. Gth,  3  times  1  are  3  ;  write  the  figure  3  a 
little  to  the  left  of  the  figure  7. 

3.  Begin  with  the  figure  2  of  the  multiplier  which 
represents  hundreds.     Hundreds  by  units  give  hun- 
dreds.    1st,   2  times   G   are    12  ;  write  the  figure   2 
under  the  column  of  hundreds  and  carry  1.     Hun- 
dreds by  tens  give  tens  of  hundreds,  or  thousands. 
2cl,  2  times  5  are  10  and  1  are  11;  write  the  figure  1 
under  the  column  of  thousands  and  carry  1.     Hun- 
dreds by  hundreds  give  tens  of  thousands.     3d,  2 
times  4  are  8  and  1  are  9 ;  write  the  figure  9  under 
the  column  of  tens  of  thousands.     In  this  case  we 
have   nothing   to  carry.      Hundreds   by   thousands 
give  hundreds  of  thousands.     4th,  2  times  3  are  G  ; 
write  the  figure  6  under  the  column  of  hundreds  of 
thousands.     In  this  case  we  have  nothing  to  carry. 
Hundreds  by  tens  of  thousands  give  ten  hundreds  of 
thousands,  or  millions.     5th,  2  times  2  are  4 ;  write 
the  figure  4  under  the  column  of  millions.     In  this 
case  also  we  have  nothing  to  carry.     Hundreds  by 
hundreds  of  thousands  give  tens  of  millions.     Gth,  2 
times  1  are  2  ;  write  the  figure  2  a  little  to  the  left 
of  the  figure  4. 

4.  Draw  a  line  under  the  several  products,  as  in 
the  example,  and  add  the  several  results  as  they  are 
found. 

The  addition  of  the  several  results  will  be  the 
answer  required.  Am.  28888704. 


58  INTERMEDIATE  ARITHMETIC. 

TO.  From  which  we  derive  the  following  general 

RULE. 

I.  Write  the  multiplier  under  the  multiplicand^  so 
that  the  units  fall  under  units,  tens  under  tens,  hun- 
dreds under  hundreds,  etc.,  etc. 

II.  Multiply  all  the  figures  of  the  multiplicand  by 
the  units  in  the  multiplier,  write  down  the  right-hand 
Jigure  and  carry  the  left-hand  one  to  be  added  to  the 
product  of  the  next  Jigure  with  the  units. 

III.  Multiply  all  the  figures  of  the  multiplicand  by 
the  figure  in  the  multiplier  representing  tens.     Write 
down  the  right-hand  figure  in  every  case  in  the  column 
of  tens  and  carry  the  left  one. 

IV.  Proceed  in  this  way  until  you  have  multiplied 
the  multiplicand  by  each  and  every  figure  of  the  mul- 
tiplier. 

V.  The  sum  of  the  several  products  will  be  the 
answer  required. 

8O.  NOTE. — When  thero  are  O's  between  the  figures  of  the 
multiplier,  pass  over  them  in  the  operation,  but  be  careful  to  give 
the  figure  to  the  left  of  the  0  its  proper  place  in  the  product. 

PEOOF. 

81.  To  prove  multiplication,  multiply  the  multi- 
plier by  the  multiplicand,  for  the  reason  that  in 
whatever  order  the  figures  may  be  multiplied  the 
result  will  be  the  same. 

EXAMPLE. 

3x2x4=24  4x2x3  =  24 

2x3x4  =  24  3x4x2  =  24 

etc.,  etc.,  etc. 


MULTIPLICATION.  59 


EXAMPLES. 


1.  Multiply  12345  by  12.  Ans.  148140. 

2.  Multiply  12345  by  123.  Ans.  1518435. 

3.  Multiply  23456  by  23.  Ans.  539488. 

4.  Multiply  23456  by  123.  Ans.  2885088. 

5.  Multiply  345678  by  1234.       Ans.  426566652. 

6.  Multiply  345678  by  12.  Ans.  4148136. 

7.  Multiply  345678  by  345.         Ans.  119258910. 

8.  Multiply  456789  by  435.         Ans.  198703215. 

9.  Multiply  456789  by  543.         Ans.  248636427. 

10.  Multiply  4576982  by  5432.  ^4^5.24862166224. 

11.  Multiply  4567289  by  12345. 

Ans.  45383182705. 

12.  Multiply  123456789  by  12.  Ans.  1481481468. 

13.  Multiply  123456789  by  123. 

Ans.  15185185047. 

14.  Multiply  123456789  by  21.  Ans.  2592592569. 

15.  Multiply  123456789  by  321. 

Ans.  39629629269. 

16.  Multiply  987654321  by  567. 

Ans.  559999990007. 

17.  Multiply  987654321  by  657. 

Ans.  648848888897. 

18.  Multiply  987654321  by  765. 

Ans.  755555554565. 

19.  Multiply  987654321  by  7065. 

Ans.  687777777786. 

20.  Multiply  8765432  by  70605. 

Ans.  618883326360. 

21.  Multiply  23546798  by  789.  Ans.  18578423622. 


60  INTERMEDIATE    ARITHMETIC. 

22.  Multiply  23546798  by  7089. 

Ans.  166923251022. 

23.  Multiply  23546798  by  70809. 

Ans.  183885219582. 

24.  Multiply  123456789  by  12304. 

Ans.  1519012331856. 

25.  Multiply  123456789  by  1034. 

Ans.  127654319826. 

26.  Multiply  12345678  by  102304. 

Ans.  1263012334856. 

27.  Multiply  23456789  by  1020304. 

Ans.  23933055643856. 

28.  Multiply  187654321  by  90809. 

Ans.  17040701235689. 

29.  Multiply  123456789  by  123456. 

Ans.  12686058042784. 

30.  Multiply  24356879  by  78901. 

.  Ans.  1921782189979. 

31.  Multiply  12387642  by  38976. 

Ans.  482820734592. 

32.  Multiply  18764321  by  12345. 

Ans.  231645542745. 

33.  Multiply  1897643  by  2468.  Ans.  4683382924. 

LESSOR  V. 

34.  Multiply  one  thousand  four  hundred  and  fifty- 
nine  b}T  twenty-three.  Ans.  33557. 

35.  Multiply    three   hundred   and    seventy-nine 
thousand  eight  hundred  and  forty-seven  by  two  hun- 
dred and  thirty-eight.  Ans.  90403586. 


MULTIPLICATION.  6 1 

36.  Multiply  three  hundred  and  forty-one  thou- 
sand nine  hundred  and  thirty-two  by  six  hundred 
and  eight.  Ans.  207894656. 

37.  Multiply  seven  hundred  and  sixty-five  thou- 
sand three  hundred  and  twenty-live  by  two  hundred 
and  eighty-five.  Ans.  218117625. 

38.  Multiply  seven  million  nine  hundred  and  for- 
ty-five thousand  three  hundred  and  one  by  two  thou- 
sand and  forty-nine.  Ans.  1C279921749. 

39.  Multiply  eighty-seven  thousand  six  hundred 
and  twenty -four  by  two  hundred  and  two. 

Ans.  17700048. 

40.  A  steam-car  travels  360  miles  a  day,  how  far 
will  it  travel  in  365  days  ?  Ans.  131400. 

41.  One  hogshead  of  sugar  weighs  462  Ibs.,  how 
many  pounds  in  263  hogsheads?  Ans.  121506. 

42.  An  army  of  29286    receives   203    dollars  as 
their  annual  pay;    what  is  the   amount   paid  the 
whole  army.  Ans.  $5945058. 

LESSON  VI. 

CASE   IV. 

7*2.  To*multiply  by  a  composite  number. 

73.  A  composite  number  is  the  product  of  two 
or  more  prime  numbers.     6  is  a  composite  number, 
because  it  is  composed  of  the  factors  2  and  3. 

2x3  =  6 

74.  A  prime  number  is  one  which  can  be  divided 
by  no  number  but  itself  and  unity.     They  are  1,  3,  5, 
7,  11,  13,  17,  19,  23,  29,  etc.,  etc. 

6 


62 


INTERMEDIATE  ARITHMETIC. 


EXAMPLE. 

Multiply  236  by  12, 

The  prime  factors  of  12  are  2,  2, 
and  3.  First  multiply  236  by  2, 
which  gives  472  for  the  product. 
Multiply  this  product  by  the  next 
factor,  2,  which  gives  944  for  the 
product.  Multiply  this  product  by 
the  other  factor  3,  which  gives  2832 
for  the  answer. 


OPEKATION. 
236 

2 

472 
2 

944 
3 


2832 


75.  This  method  is  rather  too    long.     Better 
adopt  the  rule  under  Case  III. 


EXAMPLES. 


1.  Multiply  2674  by  6. 

2.  Multiply  1358  by  12. 

3.  Multiply  8531  by  14. 

4.  Multiply  6428  by  18. 

5.  Multiply  4628  by  16. 

6.  Multiply  2648  by  9. 

7.  Multiply  8462  by  24. 

8.  Multiply  2648  by  32. 


Ans.  16044. 

Ans.  16296. 
Ans.  119434. 
Ans.  115704. 

Ans.  74048. 

Ans.  23832. 
Ans.  203088. 

Ans.  8473( 


LESSOR  VII. 

CASE    III. 

76.  To  multiply  by  10,  or  100,  or  1000,  etc.,  etc. 


MULTIPLICATION. 


63 


EXAMPLE. 


1.  Multiply  236  by  10. 

Annex  as  many  0?s  to  the  product 
as  there  are  O's  in  the  multiplier. 


OPERATION. 

236 
10 

2360 


2.  Multiply  123  by  10.  Ans.  1230. 

3.  Multiply  1234  by  10.  Ans.  12340. 

4.  Multiply  123  by  100.  Ans.  12300. 

5.  Multiply  1234  by  100.  Ans.  123400. 

6.  Multiply  12345  by  10.  Ans,  123450. 

7.  Multiply  12345  by  100.  Ans.  1234500. 

8.  Multiply  12345  by  1000.  Ans.  12345000. 

9.  Multiply  2345  by  1000.  Ans.  2345000. 

10.  Multiply  12345  by  10000.  Ans.  123450000. 

11.  Multiply  123  by  100000.  Ans.  12300000. 

12.  Multiply  1234  by  100000.  Ans.  123400000. 

13.  Multiply  1876  by  100000.  Ans.  187600000. 

14.  Multiply  1867  by  100000.  Ans.  186700000. 

15.  Multiply  12893  by  10.  Ans.  128930. 

16.  Multiply  18293  by  100.  Ans.  1829300. 

17.  Multiply  82193  by  1000.  Ans.  82193000. 

18.  Multiply  1897  by  100000.  Ans.  189700000. 

19.  Multiply  18862  by  1000.  Ans.  18862000. 

20.  Multiply  123456  by  100.  Ans.  12345600. 

21.  Multiply  1234  by  10000.  Ans.  12340000. 

22.  Multiply  123  by  10000&  Ans.  12300000. 

23.  Multiply  12  by"  1000000.  Ans.  12000000. 

24.  Multiply  123456  by  10000000,       Ans. 

25.  Multiply  654321  by  100000000.     Ans. 


64 


INTERMEDIATE   ARITHMETIC. 


LESSON  VIII. 

CASE  VI. 

7T.  In  which  the  multiplier  is  composed  of  signi- 
ficant figures  followed  by  O's. 

RULE. 

Multiply  the  multiplicand  of  the  significant  fig- 
ures and  add  to  the  result  as  many  O's  as  there  are  in 
the  multiplier. 

EXAMPLE. 

OPERATION. 


Multiply  1234  by  1600. 

1st.  Multiply  the  multiplicand  by 
16  and  to  the  product  add  two  O's. 


1234 
1600 

7344 
1234 

1968400 


1.  Multiply  1234  by  1200. 

Ans.  1480800. 

2.  Multiply  2345  by  230. 

Ans.  539350. 

3.  Multiply  2345  by  2300. 

Ans.  5393500. 

4.  Multiply  34567  by  3400. 

Ans.  117527800. 

5.  Multiply  4567  by  34000. 

Ans.  155278000. 

6.  Multiply  5678  by  4500. 

Ans.  25551000. 

7.  Multiply  6758  by  45000. 

Ans.  304110000. 

8.  Multiply  8999  by  120. 

Ans.  1079880. 

9.  Multiply  9998  by  129300. 

Ans.  1292741400. 

10.  Multiply  8764  by  3400. 

Ans.  29797600. 

11.  Multiply  7648  by  3450. 

Ans.  26385600, 

MULTIPLIC  ATI  ON .  65 

12.  Multiply  6478  by  12000.  Ans.  77736000. 

13.  Multiply  7896  by  3400.  Ans.  26846400. 

14.  Multiply  78912  by  35000.     Ans.  2761920000. 

15.  Multiply  8912  by  5600.  Ans.  49907200. 

16.  Multiply  876432  by  12000.  Ans.  10517184000. 

17.  Multiply  12345  by  1234000. 

Ans.  15233730000. 

18.  Multiply  123456  by  123450000. 

Ans.  '15240643200000. 

19.  Multiply  1357698  by  120000. 

Ans.  1(3292:3760000. 

20.  Multiply  12345678  by  13130000. 

Ans.  162098752140000. 

21.  Multiply  98989898  by  9900000. 

Ans.  979999990200000. 


LESSON  IX. 

CASE    VII. 

78.  In  which  O's  are  found  between  the  signifi- 
cant figures  of  the  multiplier  and  multiplicand. 

EULE. 

Multiply  by  the  significant  figures  only,  care  be- 
ing taken  to  give  to  each  and  every  0  its  proper  posi- 
tion amonj  the  significant  figures  in  the  product. 


66 


INTERMEDIATE   ARITHMETIC. 


EXAMPLE. 

Multiply  1023  by  203. 

1st.  Multiply  all  the  figures  in  the 
multiplicand  by  3.  Thus  3  times  3 
are  9.  3  times  2  are  6.  3  times  0 
are  0.  3  times  1  are  3. 

2nd.  Write  the  0  under  the  col- 
umn of  tens  and  multiply  by  the  next 
significant  figure  in  the  multiplier. 
Thus,  2  times  3  are  6.  2  times  2  are 
4.  2  times  0  are  0.  2  times  1  are  2. 


OPERATION. 

1023 
203 

30G9 
20460 

207669 


Add  the  several  products  for  the  answer. 


1.  Multiply 

2.  Multiply 

3.  Multiply 

4.  Multiply 

5.  Multiply 

6.  Multiply 

7.  Multiply 

8.  Multiply 

9.  Multiply 

10.  Multiply 

11.  Multiply 

12.  Multiply 


EXAMPLES. 
20304  by  201. 
302403  by  3020. 
50607  by  3040. 
60708  by  405. 
60708  by  4050. 
70809  by  5060. 
70809  by  605. 
8090  by  708. 
9080  by  8070. 
9809  by  78080. 
120980  by  70380. 
20304  by  20304. 


Ans.  4081104. 

Ans.  913257060. 

Ans.  153845280. 

Ans.  24586740. 

Ans.  245867400. 

Ans.  358293540. 

Ans.  42839445, 

Ans.  5727720. 

Ans.  73275600. 

Ans.  765886720. 

Ans.  8514572400. 

Ans.  412252416. 


MULTIPLICATION. 


67 


LESSON  X. 

CASE   VIII. 

79.  In  which  O's  are  found  after  the  significant 
figures  of  both  multiplier  and  multiplicand. 

RULE. 

Multiply  the  significant  figures,  and  add  to  the 
product  as  many  O's  as  there  are  in  the  multiplier  and 
multiplicand  together. 

EXAMPLE. 

Multiply  1200  by  1200. 

12  times  12  are  144.  Write  144 
for  your  product.  Annex  or  add  to 
it  as  many  O's  as  you  find  in  the  mul- 
tiplicand and  multiplier.  In  this  case 
four  O's.  The  answer  is  1440000. 


OPERATION 

1200 
1200 

1440000 


EXAMPLES. 


1.  Multiply  200  by  30. 

Ans.  6000. 

2.  Multiply  320  by  300. 

Ans.  96000. 

3.  Multiply  300  by  160. 

Ans.  48000. 

4.  Multiply  340  by  150. 

Ans.  51000. 

5.  Multiply  3450  by  1200. 

Ans.  4140000. 

6.  Multiply  4500  by  340. 

Ans.  1530000. 

7.  Multiply  6700  by  3400. 

Ans.  22780000. 

8.  Multiply  8900  by  450. 

Ans.  4005000. 

9.  Multiply  9800  by  4500. 

Ans.  441000000. 

10.  Multiply  7600  by  670. 

Ans.  5092000. 

11.  Multiply  5400  by  6700. 

Ans.  36180000. 

68  INTEKMEDIATE   ARITHMETIC. 

12.  Multiply  3200  by  780.  Am.  2496000. 

13.  Multiply  2100  by  7800.  Ans.  16380000. 

14.  Multiply  12000  by  8900.         Ans.  106800000. 

15.  Multiply  123000  by  9000.     Ans.  110YOOOOOO. 

16.  Multiply  12340000  by  10000. 

Ans.  123400000000. 

17.  Multiply  1234500000  by  20000. 

Ans.  24690000000000. 


CHAPTER    V. 

DIVISION. 
LESSON    I. 

80.  DIVISION  is  the  process  by  which  we  find 
how  many  times  one  number  is  contained  in  another. 

81.  The  number  by  which  we  divide  is  called 
the  divisor. 

83.  The  number  divided  is  called  the  dividend. 

83.  The  number  of  times  the  divisor  is  contained 
in  the  dividend  is  called  the  quotient. 

8  4.  If  the  dividend  does  not  contain  the  divisor 
an  exact  number  of  times,  the  excess  is  called  the 
remainder. 

EXAMPLE. 

A  boy  having  40  marbles,  wishes  to  divide  them 
equally  between  his  4  brothers.  How  many  marbles 
does  each  one  receive  ? 


DIVISION. 


G9 


ILLUSTRATION. — The  answer  is  a  number  which, 
being  multiplied  by  4,  would  give  40  for  the  product, 
or  each  boy  will  receive  as  many  marbles  as  the 
number  of  boys  (4),  is  contained  in  40,  which  is  10 
times,  because  10  times  4  are  40. 

85.  The  sign  of  division  is  a  horizontal  bar,  with 
a  dot  below  and  a  dot  above  it ;  thus,  -^-.     It  shows 
that  the  number  before  which  it  is  placed  is  to  be 
divided  by  the  one  after  it. 

86.  A  horizontal  bar  between  two  numbers,  as 
J^2,  indicates  division,  and  shows  that  12,  the  number 
above  the  line,  is  to  be  divided  by  the  one  below  the 
line.     Such  numbers  are  called  fractions. 

87.  The  remainder,  if  any,  must  always  be  less 
than  the  divisor. 


TABLE   OF  DIVISION. 


2+2=  1 

3+3=  I 

4+4=  1      5+5=  1 

6+G=  1 

7+7=  1 

4+2=  2 

6+3=  2 

8+4=  2 

10+5=  2 

12+6=  2 

14+7=  2 

6-«-2=  3 

9+3=  3 

12+4=  3 

15+5=  3 

18+6=  3 

21+7=  3 

8+2=  4 

12+3=  4 

10+4=  4 

20+5=  4 

24+6=  4 

28+7=  4 

10+2=  5 

15+3=  5 

20+4=  5 

25+5=  5 

30+6=  5 

35+7=  5 

12+2=  6 

18+3=  G 

24+4=  6 

30+5=  6 

36+6=  6 

42+7=  6 

14+2=  7 

21+3=  7 

28+4=  7 

35+5=  7 

42+6=  7 

49+7  =  7 

16+2=  8 

24+3=  8 

32+4=  8 

40+5=  8 

48+6=  8 

56+7=  8 

18+2=  9 

27+3=  9 

36+4=  9 

45+5=  9 

54  +  6=  9 

63+7=  9 

204-2=10 

30+3=10 

40+4  =  10 

50+5=10 

60+6=10 

70+7=10 

22+2=11 

33+3  =  11 

44+4  =  11 

55+5  =  11 

66+6=11 

77+7  =  11 

24+2  =  12 

36+3=12 

48+4=12 

60+5=12 

72+6=12 

84+7  =  12 

8+8=  1 

9+9=  1 

10+10=  1 

114-H=  1  :    12-r-12=  1 

16+8=  2 

18+9=  2 

20+10=  2 

224-11=  2  1     244-12=  2 

24+8=  3 

27+9=  3 

30+10=  3 

334-11=  3 

364-12=  3 

32+8=  4 

36+9=  4 

40+10=  4 

444-11=  4 

484-12=  4 

40+8=  5 

45+9^  5 

50+10=  5 

554-11=  5 

604-12=  5 

48+8=  6 

54+9=  6 

60+10=  6 

664-11=  6 

724-12=  6 

56+8=  7 

63+9=  7 

70  +  10=  7 

774-11=  7 

844-12=  7 

64+8=  8 

72+9=  8 

80+10=  8 

88+11=  8 

964-12=  8 

72+8=  9 

81+9=  9 

90+10=  9 

99+11=  9 

1084-12=  9 

80+8=10 

90+9=10 

100+10=10 

110+11=10 

1204-12=10 

88+8=11 

99+9=11 

110+10=11 

121+11  =  11 

132+12=11 

96+8=12 

108+9=12 

120+10=12 

132+11=12 

144+12=12 

70  INTERMEDIATE  ARITHMETIC. 

LESSOR  II. 

EXAMPLES. 

1.  In  120  how  many  times  10.        Ans.  12  times. 

2.  2  in  8  how  many  times  ?  in  4  ?  in  6  ?  in  12  ? 
in  14?  in  16?  in  20?  in  24? 

3.-  A  man  gave  24  dollars  to  8  boys.     How  many 
dollars  did  each  boy  receive  ?  Ans.  3. 

4.  If  10  hats  cost  40  dollars  what  costs  1  hat? 

5.  Divide  25  oranges  among  5  young  ladies.   How 
many  does  each  one  receive?  Ans.  5. 

6.  I  gave  132  dollars  for  12  chairs.     How  much 
pr.  chair  did  they  cost?  Ans.  11  dollars. 

7.  Paid  144  dollars  to  12  men.     How  much  did 
each  man  receive?  Ans.  12  dollars. 

8.  If  a  man  earn  40  dollars  in  5  weeks,  how  much 
is  that  pr.  week.  Ans.  8  dollars. 

9.  If  21   pounds   of  butter  cost   7  dollars,  how 
many  pounds  could  you  buy  for  1  dollar. 

Ans.  3  pounds. 

LESSON  III. 

EXAMPLES. 

10.  3  in  21  how  many  times  ?  in  24  ?  in  36  ? 

11.  7  in  21  how  many  times?  in  42  ?  in  56  ? 

12.  8  in  64  how  many  times?  in  16  ?  in  24? 

13.  9  in  27  how  many  times?  in  36  ?  in  45  ? 

14.  10  in  20  how  many  times  ?  in  30  ?  in  40  ? 

15.  11  in  22  how  many  times?  in  44?  in  88? 


DIVISION.  71 

J6.  12  in  24  how  many  times?  in  3C?  in  48?  in 
60? 

17.  3  in  36  how  many  times  ?  in  36  ?  in  48  ?  in 
60? 

18.  4  in  48  how  many  times?     in  36?     in  48? 
in  24? 

19.  5  in  60  how  many  times  ?     in  50?     in  25  ? 
in  45? 

20.  6  in  60  how  many  times  ?    in  48  ?     in  54  ? 
in  72? 


88.  SHORT  DIVISION, 
or  when  the  divisor  is  not  greater  than  12. 

EXAMPLE. 

Divide  8975  dollars  equally  between  5  men. 

OPERATION. 

Divisor.     5  )  8975    Dividend. 
1795     Quotient. 

We  first  find  hovr  many  times  5,  the  divisor,  is 
contained  in  8,  the  first  figure  of  the  dividend,  which 
is  thousands,  and  find  it  to  be  1  time  and  3  thou- 
sands over.  Write  the  figure  1  under  the  figure  8, 
and  carry  the  3  thousands  remaining  to  the  next 
figure  of  the  dividend.  3  thousands  and  9  hundreds 
make  39  hundreds.  Divide  39  hundreds  by  5  :  5  into 
39,  7  times  and  4  hundreds  over.  Write  the  figure 
7  under  the  figure  9,  and  carry  the  4  hundreds  to  the 
next  figure.  4  hundreds  and  7  tens  make  47  tens. 
Divide  47  tens  by  5 :  5  into  47,  9  times  and  two  tens 


72  INTERMEDIATE   ARITHMETIC. 

over.  Write  the  figure  9  under  the  figure  7,  and 
carry  the  two  tens  to  the  next  column.  2  tens  and  5 
units  make  25  units.  Divide  the  25  units  by  5 ;  5 
into  25,  5  times  and  no  remainder.  Write  the  figure 
5  under  the  5  of  the  dividend.  The  figures  thus 
written  make  the  answer.  Ans.  $1795  each. 

EULE. 

89.  I.    Write  the  Divisor  to  the  left  of  the  Divi- 
dend with  a  curved  line  separating  them. 

II.  Draw  a  horizontal  line  under  the  Dividend. 

III.  Then  beginning  at  the  left  of  the  Dividend.  Jind 
how  many  times  the  Divisor  is  contained  in  the  first 
figure  of  the  Dividend* 

IV.  If  the  first  figure  of  the  Dividend  will  not 
contain  the  Divisor,  take  the  first  two  figures,  and 
write  the  quotient  under  the  Dividend. 

V.  If  there  be  any  remainder  consider  it  as  pre- 
fixed to  the  next  figure  of  the  Dividend. 

VI.  Divide  the  number  thus  formed  by  the  Divi- 
sor and  write  the  quotient  under  the  dividend. 

VII.  If  the  number  thus  formed  will  not  contain 
the  Divisor  write  an  0  in  the  quotient  and  annex  an- 
other figure. 

VIII.  Continue  in  this  way  until  the  last  figure  is 
reached. 

IX.  When  there  is  a  remainder  write  it  a  little 
to  the  right  of  the  quotient. 

X.  It  may  be  written  in  the  form  of  a  fraction, 
the  remainder  above  the  Divisor  with  a  line  drawn 
between  them. 


DIVISION'. 


PROOF. 


OO.  FIRST  METHOD  OF  PROOF. — Multiply  the 
Quotient  by  the  Divisor  and  to  the  product  add  the 
remainder  if  any,  and  the  result  obtained  will  be  the 
Dividend,  if  the  work  is  correct. 


EXAMPLES. 


2.  Divide  8976  by  2. 

Am.  4488. 

3.  Divide  8974  by  2. 

Ans.  4487. 

4.  Divide  3993  by  3. 

Am.  1331. 

5.  Divide  319133  by  3. 

Ans.  106377|. 

6.  Divide  24684  by  4. 

Ans.  6171. 

7.  Divide  24168  by  4. 

Ans.  6042. 

8.  Divide  125125  by  5. 

Am.  25025. 

9.  Divide  125525  by  5. 

Ans.  25105. 

10.  Divide  896896  by  7. 

Ans.  128128. 

11.  Divide  986896  by  7. 

Ans.  1409854. 

12.  Divide  70011608  by  8. 

Ans.  8751451. 

13.  Divide  494580  by  12. 

Ans.  41215, 

14.  Divide  514503  by  9. 

Ans.  57167. 

15.  Divide  683409  by  9. 

Ans.  75943f. 

16.  Divide  246892  by  8 

Ans.  3086  H. 

17.  Divide  368492  by  4. 

Ans.  92123. 

18.  Divide  634829  by  4. 

Am.  158707-j. 

19.  Divide  436298  by  6. 

Ans.  72  71  6  J. 

20.  Divide  580494  by  8. 

Ans.  72561  f. 

21.  Divide  684284  by  8. 

Ans.  855351. 

22.  Divide  864824  by  9. 

Ans.  96091$. 

23.  Divide  123456  by  12. 

Ans.  10288. 

24.  Divide  123456  by  11. 

Am.  11223T\. 

25.  Divide  12345678  by  12. 

Am.  1028806}! 

74 


INTERMEDIATE   ARITHMETIC. 


LESSOR  IV, 

EXAMPLES. 

26.  Divide  23874  apples  among  12  boys.     How 
many  will  each  boy  receive.  Ans.  1989^  apples. 

27.  If  4  farthings  make  one  penny,  how  many 
pennies  in  242424  farthings.         Ans.  60606  pennies. 

28.  A  plantation  was  sold  for  2469876   dollars, 
and  the  proceeds  were  divided  equally  among  9  chil- 
dren.    How  much  was  each  child's  share. 

Ans.  $274430.66f. 

29.  Divide  24680  acres  of  land  equally  among  8 
persons.     How  many  acres  does  each  one  receive  ? 

Ans.  3085  acres. 

30.  A  school  containing  368  pupils  has  8  teachers. 
How  many  scholars  to  a  teacher?        Ans.  46  pupils. 

31.  If  3  feet  make  1  yard,  how  many  yards  in  15- 
280  feet.  Ans.  50931  yards. 

32.  If  12  pence  make  1  shilling,  how  many  shil- 
lings in  121212  pence?  Ans.  10101  shillings. 

33.  12  months  make  a  year.     How  many  years 
in  12012012  months.  Ans.  1001001  vears. 


34.  7  days  make  a  week 
77077  days. 

35.  12  articles  make  a  dozen, 
in  123456780  articles? 


How  many  weeks  in 
Ans.  11011. 


How  many  dozen 
Ans.  10288065. 


DIVISION. 


LESSON  V. 

91. — LONG   DIVISION, 

or  Division  in  which  the  Divisor  is  greater  than  12. 

EXAMPLE. 

1.  A  gentleman  having  9683425  acres  of  ground, 
planted  a  colony  of  122  persons  on  it,  and  divided 
the  land  equally  among  them.  How  many  acres 
did  each  one  receive  ? 


Write  the  divisor  on  the     j  OPERATION 

left  of   the  dividend  with  a 
curved  line  between  them. 

On  the  right  of  the  divi- 
dend draw  another  curved 
line  to  separate  it  from  the 
quotient.  Count  the  figures 
in  the  divisor  and  point  off  as 
many  in  the  dividend  begin- 
ning at  the  left.  Find  how 
many  times  the  divisor  is  con- 
tained in  the  figures  pointed 
off  in  the  dividend  and  place 
the  result  in  the  curved  line 
for  the  first  figure  of  the  quo- 
tient. If  the  corresponding 
number  of  figures  in  the  dividend  are  not  large 
enough  to  contain  the  divisor,  point  off  another  fig- 
ure. Multiply  the  divisor  by  the  quotient  found  and 
place  the  result  under  the  figures  pointed  off.  Sub- 
tract the  result  from  the  figures  pointed  off  and  take 


854 

1143 
1098 

454 
366 

882 
854 


285 
244 

41 


76 


INTERMEDIATE    ARITHMETIC. 


down  the  next  figure  of  the  dividend.  Enquire  how 
many  times  the  divisor  is  contained  in  the  new  divi- 
dend, and  write  the  result  in  the  quotient.  Multiply 
and  subtract  as  before.  If  the  remainder  together 
with  the  figure  taken  from  the  dividend  are  too  small 
to  contain  the  divisor,  write  a  cipher  (0)  in  the  quo- 
tient and  take  down  the  next  figure. 


.  The  above  example  is  read  thus:  122  into 
988,  7  times.  Multiply  the  Divisor  122  by  7  and 
write  the  product  under  the  figures  968,  and  sub- 
tract. The  remainder  is  114,  to  which  the  next  fig- 
ure (3)  of  the  Dividend  is  annexed  making  1143. 
122  in  1143,  9  times.  Multiply  the  Divisor  by  9  and 
subtract  the  product  from  1143.  45  remains  to 
which  annex  the  next  figure  (4)  of  the  Dividend, 
making  454.  122  into  454,  3  times.  Multiply  the 
Divisor  by  3  and  subtract  the  product  from  454. 
88  remains  to  which  annex  the  next  figure  (2)  of  the 
Dividend  making  882.  122  into  882,  7  times.  Mul- 
tiply the  Divisor  by  7  and  subtract  the  product  from 
882.  28  remains  to  which  annex  the  next  figure  (5) 
of  the  Dividend  making  285.  122  into  285,  2  times. 
Multiply  the  Divisor  by  2  and  subtract  the  result 
from  285  ;  41  remains.  The  Division  is  now  com- 
plete. The  answer  is  79372  and  41  remaining. 
The  answer  maybe  written  79372^^. 

03.  To  prove  the  example  multiply  the  quotient 
(79372)  by  the  divisor  (122)  and  to  the  product  add 
the  remainder  41.  The  sum  must  be  equal  to  the 
Dividend  9683425. 


DIVISION. 


77 


3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Pi  vide 
Divide 
Divide 
Divide 
Divide 
Divide 

13.  Divide 

14.  Divide 

15.  Divide 

16.  Divide 

17.  Divide 

18.  Divide 

19.  Divide 

20.  Divide 

21.  Divide 

22.  Divide 

23.  Divide 

24.  Divide 

25.  Divide 

26.  Divide 

27.  Divide 

28.  Divide 

29.  Divide 

7* 


EXAMPLES. 

24689  by  12. 
123456  by  123. 
234567  by  123. 
345678  by  123. 
435678  by  1234. 
249648  by  12. 
289246  by  123. 
184392  by  234. 
481293  by  234. 
418239  by  345. 
419293  by  45. 
196247  by  45. 
213648  by  456. 
976413  by  456. 
817924  by  56. 
213876  by  56. 
142224  by  567. 
224176  by  567. 
1869723  by  67. 
184976.  by  67. 
184976  by  678. 
247983  by  678. 
4683214  by  78. 
98764321  by  78. 
24876321  by  789. 
18764972  by  789. 
2487623  by  89. 
1876491  by  89. 
1876432  by  12345. 


Ans.  20.17,",. 
Ans.  100)5 /y.. 
Ans.  19" 
Ans. 
Ans. 

Ans.  20804. 

Ans.  2351^. 

Ans.  788. 

Ans.  2056fff. 

Ans.  1212jftfr. 

Ans.  93l7|f. 

Ans.  4::< 

Ans.  4G8HJ. 

Ans. 

Ans. 

Ans.  3819£f. 

Ans.  250-H4. 

Ans.  395-f^-. 

Ans.  2789|f 

Ans.  2760j-fT 

Ans.  274;;!!;. 

Ans.  365fL|. 

Ans.  60041f«-. 

Ans.  12662094$. 

Ans.  31528-^f  f. 

Ans.  23780^5. 

Ans.  27838f|. 

Ans.  21084-if. 

Ans. 


78  INTERMEDIATE  AEITHMETIC. 

LESSOR  VI. 

32.  Asia  has  a  population  of  696725000.     Louis- 
iana has  a  population  of  708002.     How  many  times 
more  population  has  Asia  than  the  State  of  Louis- 
iana? Am.  984^y>o¥- 

33.  Australia  has  a  population  of  1431000.  Rhode 
Island,  174621.     How  many  times  greater  is  the  pop- 
ulation of  Australia  than  that  of  Rhode  Island  ? 

Ans.  s^fJffifr  times- 

34.  Louisiana  has  an  area  of  46431  square  miles. 
If  it  was  divided  equally  among  the  48  parishes  in 
the  State,  what  would  be  the  area  of  each  parish  ? 

Ans.  96 7J-£  square  miles. 

35.  New  York  State  has  3,880,735  inhabitants, 
and  an  area  of  47000  square  miles.     How  many  in- 
habitants is  that  to  a  square  mile  ?       Ans.  82f -f-Jf f . 

36.  The  Middle  States  have  7571099  inhabitants, 
and  an  area  of  103440  square  miles.     How  many  in- 
habitants to  a  square  mile  ?  Ans.  73^°^%. 

37.  The  Southern  States,  11  in  number,  have»a 
population  of  8321301   inhabitants   and  an  area  of 
644551  square  miles.     If  the  states  were  of  the  same 
area,  how  many  square  miles  would  each  one  contain  ? 

If  the  inhabitants  were  equally  divided  between 
them,  how  many  inhabitants  would  each  State  con- 
tain ? 

How  many  inhabitants  to  a  square  mile  in  the 
Southern  States? 

Ans.  58595^         square  miles  to  each  State. 
75648TftT         inhabitants  to  each  State. 
102-jj-fj-jHrf-  inhabitants  to  each  sq.  mile. 


DIVISION.  70 

38.  Belgium  has  a  population  of  4731957  inhab- 
itants and  an  area  of  11313  square  miles.    How  many 
inhabitants  to  a  square  mile  ?  Ans. 

39.  The   Eastern   States   have    a   population   of 
3135282   inhabitants,  and  an  area  of  G5038   square 
miles.     How  many  inhabitants  to  a  square  mile  ? 

Am. 


40.  A  colony  of  248  persons  purchased  1864358 
acres  of  land.     How  many  acres  to  each  person  •? 

Ans.  7  5  17  \\l. 

41.  5280  feet  make  one  mile.     3  feet  make  1  yard. 
How  many  yards  in  a  mile  ?  Ans.  1760. 

42.  63360  inches  make  1  mile.     12  inches  make  1 
foot.     How  many  feet  in  a  mile?  Ans.  5280. 

43.  12  articles  make  1  dozen.     How  many  dozen 
in  1728  articles?  Ans.  144. 

45.  A  gentleman  divided  567053  cents  among  24 
beggars.     How  much  did  each  beggar  receive  ? 

Ans.  2362  if  cents. 

46.  A  teacher  buys  123456789  marbles.     In  his 
school  there  are  348  boys.     How  many  marbles  could 
he  give  to  each  one  of  the  boys  ? 

Supppose  -I-  of  the  boys  were  absent,  how  many 
would  those  who  were  present  get  ? 

1st  Ans.  35476|ff. 
2d  Ans. 


80 


IXTEKMEDIATE    ARITHMETIC. 


LESSON  VII. 
©4L  To  Divide  by  a  composite  number. 

EXAMPLES. 

I.  Divide  240  by  12. 

12  is  composed  of  the  factors  3X 
2X2.  Divide  240  by  3.  The  quo- 
tient is  80.  Divide  80  by  2.  The 
quotient  is  40.  Divide  40  by  2.  The 
quotient  is  20.  The  last  quotient  (20) 
is  the  answer. 

2.  Divide  1230  by  6. 

3.  Divide  2340  by  4. 

4.  Divide  23460  by  4. 

5.  Divide  1800  by  9. 
G.  Divide  3600  by  18. 

7.  Divide  1212  by  4. 

8.  Divide  2424  by  8. 

9.  Divide  36036  by  4. 
10.  Divide  36036  by  9. 

II.  Divide  1240  by  4. 

12.  Divide  1240  by  8. 

13.  Divide  4500  by  15. 

14.  Divide  8888  by  8. 

15.  Divide  6464  by  64. 

16.  Divide  12012  by  6. 

17.  Divide  3648  by  6. 

18.  Divide  3600  by  12. 

19.  Divide  8400  by  14. 

20.  Divide  8000  by  16.   <• 

21.  Divide  1620  by  18. 

22.  Divide  24200  by  21. 


OPEKATIOS. 

3)240 
2)80" 
2)40 
Ans.  20 

Ans.  205. 

Ans.  585. 
Ans.  5865. 

Ans.  200. 

Ans.  200. 

Ans.  303. 

Ans.  303. 
Ans.  9009. 
Ans.  4004. 

Ans.  310. 

Ans.  155. 

Ans.  300. 
Ans.  1111. 

Ans.  101. 
Ans.  2002. 

Ans.  608. 

Ans.  300. 

Ans.  600. 

Ans.  500. 
Ans.  90. 
Ans.  1152/T. 


DIVISION.  81 

9«>.  To  find  the  true  remainder  : 
RULE. 

Multiply  all  the  remainders  by  all  the  divisors 
except  its  own,  beginning  at  the  last  and  multiplying 
upwards  /  add  the  first  remainder  to  the  sum  of  the 
products,  and  the  amount  will  be  the  true  remainder. 

(Since  every  remainder  is  multiplied  by  all  the 
divisors  from  the  one  which  produced  it,  the  first 
remainder  will  have  no  multiplier.) 

EXAMPLE. 

Divide  798  by  44. 


The   factors   of  44   are 


2X2X11.     Dividing   by  2  g 

gives  299  and  no  remainder. 


Dividing  the  quotient  by  2 

a  „            .   -                 11)199—1     2d  rem. 
gives  199  and  2  remaining.  I 

Dividing  the  quotient  by 
11  gives  18  and  1  remain- 
ing. 

To  find  the  true  remain- 
der, begin  with  the  last  re- 
mainder, 1,  and  multiply  it 
by  all  the  divisors  except  its  own ;  the  result  will  be 
4.  Multiply  the  second  remainder  by  all  the  divisors 
except  its  own ;  the  result  will  be  2.  Adding  the 
results  gives  6,  to  which  ^add  the  first  remainder,  0. 
The  amount  is  6,  which  is  the  true  remainder. 

The  answer  of  the  sum  is  18  and  6  remaining. 


OPERATION. 


6 


82 


IJSTTERMEDIATE   ARITHMETIC. 


PEOOF. 

44  )  798  (  18  and  6  remaining. 
44 

358 
352 


G 


LESSON  VIII. 
OG.  When  the  divisor  ends  in  O's. 

EXAMPLE. 

1.  Divide  2468  by  10. 


OPEKATIOtf. 


10)24G|8 


246 — 8  rem. 


"VVe  know  that  to  multiply 
any  number  by  10,  an  0  annexed 
to  the  multiplicand  will  give 
the  required  product.  Division 
is  the  reverse  of  multiplication.  If  we  wish  to  divide 
by  10,  we  must  cut  off  one  figure  from  the  right  of 
the  dividend.  The  remaining  figures  on  the  left  will 
be  the  quotient,  and  the  figure  cut  off  will  be  the  re- 
mainder. The  answer  in  the  above  example  is  246 
and  8  remaining. 

To  divide  by  100,  1000,  etc.,  etc.,  cut  off  as  many 
figures  from  the  right  of  the  dividend  as  there  are  O's 
in  the  divisor.  The  figures  to  the  left  of  the  line  is 
the  quotient,  those  on  the  right  the  remainder. 

2.  Divide  12345  by  10. 

Ans.  1234  and  5  remaining. 


DIVISION.  83 

3.  Divide  23456  by  100. 

Am.  234  and  56  remaining. 

4.  Divide  34567  by  1000. 

Ans.  34  and  567  remaining. 

5.  Divide  45G78  by  10000. 

Ans.  4  and  5678  remaining. 

6.  Divide  987654321  by  1000000. 

Ans.  987  and  654321  remaining. 

7.  Divide  123456789  by  1000000. 

Ans.  123  and  456789  remaining. 


LESSON"  IX. 

97.  To  buy  or  sell  articles  by  the  hundreds, 
thousands,  etc. 

EXAMPLE. 

1.  What  will    1234567    bricks    cost  @    $12  per 
thousand. 

By  the  thousand  means  for  every     j      OPEEATI0*- 
1000.     To  find  the  number  of  thou- 
sands  in  any  number,   cut   off  as 
many  figures  from  the  dividend  as 
there  are  O's  in  1000.     Multiply  the  2468734 

number  by  the  price,  care  being  ta-  1234567 

ken  to  cut  off  as  many  figures  from 
the  result  as  there  are  figures  cut  off 
in  the  multiplicand.     The  figures  to  the  left  of  the 
line  in  the  result  are  dollars,  cents,  etc.,  as  the  case 
may  be.    Those-  to  the  right  of  the  line  are  fractional 
parts  of  dollars,  etc. 


84 


INTERMEDIATE    ARITHMETIC. 


In  the  following  examples  we  will  treat  of  dollars 
only  in  the  multiplier. 

EXAMPLES. 

2.  What  will  12345  bricks  cost  @  $16  per  thou- 
sand ?  Ans.  $197j520. 

3.  What  will  234567  slates  cost  @  $25  per  thou- 
sand? Ans.  $5 864 j  175. 

4.  What  will  267433  feet  of  lumber  cost  @  $22 
per  thousand?  Ans.  85883(526. 

5.  What  will  34567  bricks  cost  @  $14  per  thou- 
sand? Ans.  8483)938. 

6.  What  will  123456  shingles  cost  @  $2  per  hun- 
dred? Ans.  $246 1 90. 

7.  What    will   654321   shingles   cost  @  $3  per 
hundred?  Ans.  $19629163 

8.  What  will  8765432  bricks  cost  @  $2  per  hun- 
dred? Ans.  $!75308i64. 

9.  What  will  2345678  bricks  cost    @   $20  per 
thousand?  Ans.  $469131560. 

10.  What  will  9876543   slates  cost   @   $20   p< 
thousand  ?  Ans. 

11.  What  will  3456789  slates  cost  @  $2  per  hun- 
dred ?  Ans. 

12.  What  will  5464748  bricks  cost  @  $3  per  hun- 
dred ?  Ans. 

13.  What  will  8474645  bricks  cost  @  $30  per  thou- 
sand ?  Ans. 

14.  What  will  123456789  feet  of  lumber  cost  @ 
$14  per  thousand  ?  Ans. 


DIVISION.  85 

LESSON  X. 

98.  To  divide  without  writing  the  result  under 
the  dividend  ;  that  is,  to  subtract  mentally  the  result 
from  the  dividend. 

EXAMPLE. 

Divide  934  by  33. 

33   into  93,  2   times.     Write  the  OPERATION. 

figure  2  in  the  quotient,  and  multiply.          33)934(23 
2  times  3  are  6.     Instead  of  writing  974 

the  figure  6  under  the  figure  3  of  the 
dividend,  subtract  it  from  the  figure 
3,  and  write  the  difference  under  the  dividend ;  thus : 
2  times  3  are  6 ;  6  from  3  I  cannot,  borrow  10 ;  6 
from  13  leaves  7.     2  times  3  are  6.    To  the  figure  G  I 
must  now  add  the  1  ten  I  borrowed.     6  and  1  are  7. 
7  from  9  leaves  2.     Write  the  figure  2  under  the  fig- 
ure 9.     The  remainder  is  27.     Take  down  the  next 
figure  (4)  of  the  dividend.     The  new  dividend  is  274. 

33  into  274,  8  times.  8  times  3  are  24.  4  from  4 
leaves  0.  Write  the  figure  0  under  the  figure  4.  8 
times  3  are  24  and  2  are  26.  26  from  27  leaves  1. 
Write  the  figure  1  under  the  figure  7.  The  answer 
is  28  and  10  remaining. 

PROOF. 

33     Divisor. 
28     Quotient. 


264 
66 

924     Product. 
10    Remainder. 

934    Dividend. 


86  IHTTEEMEDIATE  AEITHMETIC. 


EXAMPLES. 

1.  Divide  6789  by  15.   Ans.  452  and  9  remaining. 

2.  Divide  7892  by  14. 

Ans.  563  and  10  remaining. 

3.  Divide  7829  by  13.   Ans.  602  and  3  remaining. 

4.  Divide  2234  by  12.    Ans.  186  and  2  remaining. 

5.  Divide  45678  by  16. 

Ans.  2854  and  14  remaining. 

6.  Divide  56478  by  17. 

Ans.  3322  and  4  remaining. 

7.  Divide  78645  by  18. 

Ans.  4369  and  3  remaining. 

8.  Divide  165487  by  19. 

Ans.  8709  and  16  remaining. 

9.  Divide  76432  by  21. 

Ans.  3639  and  13  remaining. 

10.  Divide  89765  by  24. 

Ans.  3740  and  5  remaining. 

11.  Divide  24897  by  28. 

Ans.  889  and  5  remaining. 

12.  Divide  389764  by  31. 

Ans.  12573  and  1  remaining. 

13.  Divide  983764  by  211. 

Ans.  4662  and  82  remaining. 

14.  Divide  876432  by  321. 

Ans.  2730  and  102  remaining. 

15.  Divide  8976421  by  461. 

Ans.  19471  and  290  remaining. 

16.  Divide  978642  by  112. 

Ans.  8737  and  98  remaining. 


DIVISION.  87 

17.  Divide  8796431  by  213. 

Ans.  41297  and  170  remaining. 

18.  Divide  8764921  by  318. 

Ans.  27562  and  205  remaining. 

19.  Divide  31314151  by  541. 

Ans.  39397  and  374  remaining. 

20.  Divide  12345678  by  1234. 

Ans.  10004  and  742  remaining. 

21.  Divide  23456789  by  2345. 

Ans.  10002  and  2099  remaining. 

22.  Divide  98765423  by  1345. 

Ans.  73431  and  728  remaining. 

23.  Divide  34567898  by  1456. 

Ans.  23741  and  1002  remaining. 

24.  Divide  123456789  by  1567. 

Ans.  78785  and  649  remaining. 

25.  Divide  98234765  by  1678. 

Ans.  58548  and  721  remaining. 

26.  Divide  82934569  by  2135. 

Ans.  38845  and  494  remaining. 

27.  Divide  98276431  by  2146. 

Ans.  45795  and  361  remaining. 

28.  Divide  897643218  by  2133. 

Ans.  420836  and  30  remaining. 

29.  Divide  876432182  by  8697. 

Ans.  100774  and  704  remaining. 

30.  Divide  764328973  by  213456. 

Ans.  3580  and  156293  remaining. 


88  INTERMEDIATE   ARITHMETIC. 

LESSOR  XL 

99.    EXAMPLES   IN  NUMERATION,  NOTATION,  AND  THE 
FOUR   BULES. 

Write  in  figures  the  following  : 

1.  Six   hundred   and    ninety-six   millions    seven 
hundred  and  twenty-five  thousand  nine  hundred  and 
eighty-one. 

2.  Four  millions  three  hundred  and   eighty-six 
thousand  four  hundred  and  ninety-seven. 

3.  Four  hundred  and  eleven  billions  one  hundred 
and  fourteen  millions   one   hundred   and   forty-one 
thousand  one  hundred  and  fourteen. 

4.  Two  hundred  and  eighty  millions  nine  hundred 
and  fifty-nine  thousand  five  hundred  and  five. 

5.  Three   millions   five   hundred   and    forty-four 
thousand  eight  hundred  and  seventy-seven. 

6.  Two  hundred  and  forty-three  trillions   three 
hundred  and  twenty- four  billions  one  hundred  and 
sixty-two  millions  two  thousand  one  hundred  and 
one. 

7.  Three    trillions  three   billions   three   millions 
thirty  thousand  and  three  hundred. 

8.  One  quadrillion  one  thousand. 

Read  the  following  numbers: 

9.         123456789. 

10.  1234567890. 

11.  12345678901. 

12.  123456789012. 


MISCELLANEOUS  EXAMPLES.  89 

13.  12345987643. 

14.  2345987643. 

15.  439587643. 

16.  87642182.  . 

17.  1876431. 

18.  879643S. 

19.  187643. 

20.  278692. 

21.  83217. 

22.  9462. 

23.  873. 

24.  123456789101234552. 

LESSON"  XIT. 

25.  The  following  is  the  population  of  the  6  New 
England  States:  Maine,  628279;  Vermont,  315098; 
New  Hampshire,  326072 ;   Massachusetts,  1231065; 
Connecticut,  460147;  Rhode  Island,  174621.     How 
many  inhabitants  in  the  New  England  States  ?    How 
many  more  has  Massachusetts  than  Rhode  Island. 
Which  is  the  greater,  Maine  or  Vermont,  and  by  how 
many?     How  many  more  has  Massachusetts  than 
Rhode   Island   and   Connecticut    together?      Than 
Maine   arid   Vermont   together?     If   Massachusetts 
was  taken  from  the  Eastern  States,  how  many  inhab- 
itants would  there  be  in  the  remaining  States  ? 

First  Ans.  3135282. 

26.  The  population  of  the  4  Middle  States  is  as 
follows:  New  York,  3880735;  Pennsylvania,  2906- 
115 ;  New  Jersey,  672031 ;  Delaware,  112218.    How 

8* 


90 


INTERMEDIATE   ARITHMETIC. 


many  inhabitants  in  the  Middle  States  ?  How  many 
more  has  New  York  than  Delaware  ?  How  many 
more  has  New  York  than  New  Jersey  and  Delaware 
together?  Has  New  York  more  than  Delaware  and 
Pennsylvania  both  ?  If  so,  how  many  ?  How  many 
times  greater  is  the  population  of  New  York  than 
New  Jersey  ?  First  Ans.  7571099. 

27.  The  population  of  the  Southern  States  is  dis- 
tributed as  follows:  Texas,  604215  ;  Virginia,  1219- 
630 ;  West  Virginia,  376688 ;  Florida,  140425  ;  Geor- 
gia, 1057286  ;  Alabama,  964201 ;  Mississippi,  791395  ; 
Louisianaj  708002 ;  North  Carolina,  992622  ;  South 
Carolina,  703708;    Maryland,  687049;    District  of 
Columbia,   75080.     How   many   inhabitants  in   the 
Southern  States  ?     Which  has  the  most  population, 
West  Virginia  or  the  District  of  Columbia  ?     Vir- 
ginia or  West  Virginia?     Louisiana  or  Maryland? 
Maryland  or  Georgia?  and  by  how  many?     How 
many  times  more  inhabitants  has  Georgia  than  the 
District  of  Columbia  ?    How  many  more  inhabitants 
have  the  Southern  States  than  the  Middle  States  ? 

Last  Ans.  749202. 

28.  The  population  of  the  Western  States  is  dis- 
tributed as   follows:    California,    380015;    Oregon, 
52465;   Minnesota,   173885;    Kansas,   107208;   Mis- 
souri, 1182014;  Michigan,  749112;  Illinois,  1711951 ; 
Wisconsin,   775881;  Arkansas,   435450;  Iowa,  674- 
948  ;  Tennessee,  1109801 ;  Ohio,  2339511 ;  Kentucky, 
1155684;   Indiana,    1350428;   Nevada,  6857.     How 
many  inhabitants  in    the  Western  States?    How 
many  more  in  the  Western  than  in  the  Southern 


MISCELLANEOUS   EXAMPLES.  91 

States ?    How  many  more  has  Ohio  than  Louisiana? 
How  many  times  more  has  Ohio  than  Nevada  ? 

First  Ans.  1218G210. 

29.  The  population  of  the  territories  is  divided  as 
follows:  Dakota,  4839;  New  Mexico,  93541 ;  Wash- 
ington, 11578;  Utah,  40295;  Nebraska,  28842;  Col- 
orado, 34197;  Indian,  97G1.     How  many  inhabitants 
in  the  above-named  territories  ?     How  many  inhab- 
itants has  Louisiana  more  than  all  the  above  terri- 
tories ?     How  many  times  more  inhabitants  have  the 
Western  States  than  all  of  the  territories? 

Last  Ans.  54-f. 

30.  Russia  has  2100000  square  miles;  Louisiana, 
46431.    How  many   times   greater  is   Russia  than 
Louisiana? 

31.  The  area  of  the  -United  States  is   1147485 
square  miles;    Great  Britain  and  Ireland,   122550. 
How  many  times  larger  is  the  United  States  than 
Great  Britain  and  Ireland?  Ans.  9-}-. 

32.  I  buy  from  John  Smith  73  hogsheads  @  22 
dollars  per  hogshead,  and  sold  them  for  25  dollars 
per  hogshead.     Do  I  gain  or  lose,  and  how  much  ? 

Ans.  Gains  219  dollars. 

33.  J.  Brown  owes  to  one  man  286  dollars,  to 
another  he  owes  798  dollars,  to  a  third  he  owes  685 
dollars.     John   Smith    owes   him   3452  dollars.     If 
John  Smith  pays  him,  and  he  pays  his  debts,  how 
much  will  he  have  left  ?  Ans.  $1683. 

34.  A  butcher  bought  2  cows,  eight  hundred  and 
forty-nine  pounds  each,  at  11  cents  a  pound,  and  sold 
them  @  12  cents  a  pound.     How  much  did  he  gain? 

Ans.  $16.98. 


92  INTERMEDIATE   ARITHMETIC. 

35.  The  butcher  had  four  partners.     "What  was 
the  share  of  each  partner  ?  Ans.  $4.24. 

36.  The  wheel  of  my  car  is  4  feet  in  circumfer- 
ence.    If  I  travel  5280  feet,  how  many  times  will  the 
wheel  have  turned?  Ans.  1320  times. 

37.  The  wheel  of  my  sulky  is  4  feet  in  circumfer- 
ence.    In  one  of  my  recent  drives,  the  wheel  turned 
22640  times.     How  many  miles  did  I  drive,  allowing 
5280  feet  to  the  mile  ?  Ans.  17  miles. 

38.  I  sold  180  spools  thread  @  20  cents,  10  yards 
calico   @    15    cents,  46  gallons  oil  @  28  cents,  17 
pounds   tea   @  46  cents,   102   pounds  coffee   @   32 
cents.     How  much  was  my  bill  ?     (100  cents  make  a 
dollar.)  Ans.  $90.84. 


LESSON  XIII. 

39.  The  distance  from  New  York  to  Canton  is  li 
450  miles.     How  many  days  would  a  steamer  require 
to  reach  Canton  traveling  at  the  rate  of  200  miles  a 
day?  Ans.  97+. 

40.  Bought  of  Sam.  Johnson,  200  acres  of  land  @, 
$3  per  acre,  and  paid  him  as  follows ;  2  ponies  valued 
@  $75  each,  2  small  cottages  valued  @  $100  each,  12 
thousand  feet  of  lumber  @  $12.     How  much  do  I  owe 
him  yet  ? 

41.  Four  boys  find  a  mulberry  tree  and  agree  to 
pick  in  partnership.     In  one  hour  they  picked  up  18- 
964.     How  many  should  each  boy  receive  as  his 
share  ?  Ans.  4741. 


MISCELLANEOUS   EXAMPLES.  93 

42.  A  hotel  cost  186897  dollars  to  build  it.     The 
expense  was  equally  borne  by  18  men.     How  much 
did  each  man  have  to  pay?  Am.  10383 -f-. 

43.  The  building   of  the  Chattanooga  Railroad 
will  cost  about  ten  million  of  dollars.     There  are  em- 
ployed on  the  road  about  2804  men.     How  much  is 
that  to  each  man  ?  Ans.  3491  -}-. 

44.  What  is  the  amount  of  the  following  bill  ?  200 
boxes  soap  @  $3,  50  kegs  lard  @  $12,  25  boxes  can- 
dies  @  $8,  4  bbls.  sugar  @  812.      If  I  gave  the  gro- 
cer three  five-hundred  dollar  notes,  how  much  change 
should  I  get  back  ?  Ans.  $52. 

45.  Mt.  St.  Elias  is  IV 800  feet  high.     How  long 
would  it  take  to  climb  to  the  top,  at  the  rate  of  3800 
feet  per  hour  ?  Ans.  4  hrs.  42  min. 

40.  Jamestown,  Va.,  was  settled  in  1607.     How 
old  is  Jamestown  ?  Ans.  262  years  old. 

47.  In  the  city  of  London  there  are  2803034  in- 
habitants.    How  many  regiments  of  980  men  each, 
could  London  furnish,  providing  the  inhabitants  were 
all  men?  Ans.  2860-f-  reg. 

48.  Mt.  Everest  is  twenty-nine  thousand  and  two 
feet  in  height.     Mt.  St.  Elias  is  seventeen  thousand 
eight  hundred  and  sixty.     How  much  higher  is  Mt. 
Everest  than  Mt.  St.  Elias  ?  Ans.  11142. 

49.  Lake  Sirikol  is  15600  above  the  level  of  the 
sea.     The  Dead  Sea  is  1312  feet  below  the  level  of 
the  sea.     How  high  is  Lake  Sirikol  above  the  Dead 
Sea?  Ans.  16912. 

50.  Lake  Itasca  is  one  thousand  six  hundred  and 
seventy-five  feet  above  the  level  of  the  sea.     The 


94 


INTERMEDIATE 


METIC. 


Caspian  Sea  is  eighty-three -feet  below  the  tevel  of 
the  sea  ?     How  much  higjjj(pis  Lake  Itasca  ? 

Ans.  1658. 

51.  Washington  died.in_l  7 9 9.    La  Fayette  visited 
America   in  1824.,    How   long   after   Washington's 
death?  ^  Ans.  25  years. 

52.  Make  the  foiling  bill: 


2  doz.  slates 
2  boxes  chalk 
4  inkstands 
8  doz.  penholders 
4  qrs.  paper 


20  cts.  apiece. 
75  cts.       " 
12  cts.      '" 
1  ct.        " 
25  cts.       " 


5,74 


53.  Make  the  following  bill  : 

2  pr.  shoes      @     $4     . 


"   boots      @ 
"   brogans  @ 


$8 
$3 


$71.00 


54.  If  I  give  the  merchant  $150,  how  much  must 
I  get  back  ?  Ans.  $79.00 

f>5»  What  is  the  amount  of  ttie  following  bill: 

Lpz.  grammars     @  $2  each  .     .  . 

readers          @  $1  each  .     .  . 

9    "     spellers         @  $1  each  .     .  . 

7    "    geographies  @  $3  each  ...  . 


Amount 


$480 


QUESTIONS   INVOLVING  FR ACTIONS.  95 

56^plf  I  give  the  merchant  $600,  how  much  must 
he  give  me  in  return ?  -I//*.  $120. 

57.  The  building  of  a  railroad  cost  987654321  dol- 
lars. There  were  864321  shares  sold  to  realize  that 
sum.  How  much  is  each  share  worth  ?  If  the  road 
cost  $55000  per:  mile,  how  long  would  it  be  ? 

First  Ans.  1142.69+. 


CHAPTER    VI. 

QUESTIONS  INVOLVING  FRACTIONS. 
LESSON    I. 

'1OO.  If  a  unit  or  a  single  thing  is  divided  into 
parts,  each  part  is  a  fraction  of  that  unit;  or,  better, 

101.  A  fraction  is  a  part  of  a  unit. 

102.  If  an  apple  is  cut  into  2  parts,  each  part  is 
equal  to  one-half  of  the  apple.     One-half  is  written 
(!)  with  the  1  above  the  figure  2,  and  a  line  drawn 
between  them. 

103.  The  figure  below  the.  line  is  called  the 
denominator,  because  it  shows  into  how  many  parts 
the  unit  is  divided. 

1O4L  The  figure  above  the  line  is  called  the 
numerator^  and  shows  how  many  of  those  parts  into 
which  the  unit  is  divided,  are  used.  Thus,  £  shows 
that  the  unit  is  divided  into  8  equal  parts,  and  that  5 
of  these  parts  are  taken. 


96  INTERMEDIATE  ARITHMETIC. 


If  a  unit  is  divided  into  thirds,  fourths, 
fifths,  sixths,  etc.,  and  one  of  the  equal  parts  is  taken, 
the  fraction  is  written  thus  :  -J,  |,  -J,  -J-,  etc. 

106.  When  the  numerator  is  equal  to  the  de- 
nominator, the  fraction  represents   a  whole  unit  or 
thing;  because,  if  an  object  is  divided  into  10  equal 
parts,  and  10  of  those  parts  are  taken,  the  whole  unit 
is  taken. 

107.  If  a  unit  is  divided  into  4  equal  parts,  and 

2  of  those  parts  are  taken,  there  remains  2  parts,  or 
f  ;  because,  if  a  unit  is  divided  into  4,  £  is  equal  to 
the  unit,  and  if  2  fourths  are  taken  away,  there  re- 
mains f-. 

EXAMPLES. 

1.  If  a  unit  is  divided  into  14  equal  parts,  what  is 

3  of  those  parts  called?     6  parts?     8  parts? 

2.  Write   in   figures  three-fourteenths,   six-four- 
teenths, eight-fourteenths. 

108.  To  find  the  half,  third,  fourth,  etc..  of  any 
number,  divide  the  number  by  2,  3,  4,  etc. 

EXAMPLES. 

3.  What  is  one-half  of  14?    16?    18?    20?    22? 
24?     26?     28?     30? 

4.  What  is  one-third  of  3?     6?     9?     12?     15? 
18?     21?     24?     27?     30?     33? 

5.  What  is  one-fourth  of  4  ?    8?    12?    16?     20? 
24?     28?     32?     36?     40? 

6.  What  is  one-fifth  of  5  ?     15  ?     25  ?     35  ? 


QUESTIONS   INVOLVING   FRACTIONS.  9V 

LESSON  II. 

1O9.  When  several  of  the  equal  parts  are  taken. 

EXAMPLE. 

7.  What  is  |  of  C  ? 

SOLUTION. 

2  x  6  =  12  ^=4,  the  answer. 

KULE. 

Multiply  the  whole  number  by  the  numerator  and 
divide  the  result  by  the  denominator. 

EXAMPLES. 

8.  What  is  |  of  4?     8?     12?     16?     20?     24? 

9.  What  is  |  of  5?    10?     15?     20?     25?     30? 

10.  What  is  |  of  6  ?    12?     18?     24?     30?     36? 

11.  What  is  -\  of  7?    14?     21?     28?     35?     42? 

12.  What  is  f  of  42  ?     84  ?     168  ?     336  ?     672  ? 

13.  What  is  |  of  8  ?    16  ?  24  ?  32  ?  40  ?  48  ?  56  ? 

14.  What  is  f  of  112  ?     224?     448?     896? 

15.  What  is  f  of  9?    81?    27?     36?     45?     90? 

16.  James  Brown  gave  -J-  of  his  cake  to  his  sister 
and  -J  to  his  brother.     What  part  of  the  cake  did  he 
keep  for  himself  ? 

SOLUTION. 

•J-  and  -J  are  |,  the  part  given  away, 
f  make  one  whole  cake.     From  f  take  -f ,  and 
there  remains  (5  from  8)  -|. 
9 


98  INTERMEDIATE   ARITHMETIC. 

17.  From  one  whole  one  take  -|,  f ,  -|. 

18.  From  a  load  of  pumpkins  I  gave  Jones  f. 
How  many  ninths  had  I  remaining.  ? 

19.  In  a  school,  -f^  study  grammar;  -^  geogra- 
phy; y4,,  history;  and  the  balance  study  rhetoric. 
What  part  of  the  school  study  rhetoric  ? 

20.  From  a  hogshead  of  sugar  I  sold  ^.     How 
many  elevenths  remained  ? 

21.  If  1  yard  of  cloth  costs  10  dollars,  what  will 
£  a  yard  cost  ? 

SOLUTION. 

If  1  yard  costs  10  dollars,  £  yard  will  cost  -|-  of  10 
dollars,  or  5  dollars. 

22.  If  1  acre  of  land  costs  $40,  what  will  £  of  an 
acre  cost  ?    f  of  an  acre  cost  ? 

23.  If  1  acre  of  land  costs  $240,  what  will  £  of  an 
acre  cost  ?    f  cost  ?    f  cost  ?    f  cost  ? 


LESSON  III. 

24.  If  -J-  of  a  hogshead  of  sugar  is  worth  120  dol- 
lars, what  is  i  worth  ?  ^  worth  ? 

SOLUTION. 

Since  |  is  equal  to  120  dollars,  £  is  equal  to  -J-  of 
120  which  is  24.     If  |  is  equal  to  24,  4  is  equal  to  96. 

25.  If  |  of  an  acre  of  land  is  worth  $210,  how 
much  is  %  worth  ?  f?  |  ?  |  ? 


QUESTIONS   INVOLVING   FRACTIONS.  99 

26.  If  f  of  a  farm  is  worth  $1000,  how  much  is 
|  worth?  f?  }?  |?  |? 

27.  If  ft  of  a  farm  is  worth  9000  dollars,  how 
muc-h  is  TV  worth  ?  ft  ?  ^  ?   ft  ?  ft  ?  ft-  ?  ft  ? 


LESSON  IV. 

28.  How  many  half  pounds  in  4  pounds  and  £  of 

a  pound  ? 

SOLUTION. 

In  1  pound  there  are  2  halves.  In  4  pounds  there 
are  4x2  halves  or  8  halves.  In  4  and  1  half  pounds 
there  would  be  8  halves  and  1  half  or  9  halves  f . 

Or,  better : 

4£  is  called  a  mixed  number.  A  mixed  number 
consists  of  a  whole  number  and  a  fraction-,  f  is  an 
improper  fraction.  To  reduce  a  mixed  number  to  an 
improper  fraction : 

Multiply  the  whole  number  by  the  denominator  of 
the  fraction,  and  to  the  sum  add  the  numerator.  Un- 
der the  result  thus  found  write  the  denominator. 

• 

29.  How  many  thirds  in  6f.  Ans.  %£-. 

SOLUTIOX. 

[(6  X  3)-f-2]-^3      6x3  =  18     18  +  2  =  20      20^-3=^. 

30.  How  many  thirds  of  a  dollar  in  3|?  4£  ?  5|  ? 
G-|?  ?f  ?  8J?  9|?  10J-? 

31.  How  many  fourths  of  an  ounce  in  l£?  2f? 
3f?  4£?  5J?  6f?  7|?  81?  9f?  10J? 


100  INTERMEDIATE   ARITHMETIC. 


LESSON  V. 

32.  How  many  boxes  cheese  in  18  half  boxes? 

20?  22?  24?  26?  28?  30?   32?  34?  36?  38?  40? 
Why  ? 

The  above  is  reducing  an  improper  fraction  to  a 
mixed  number.  To  reduce  an  improper  fraction  to 
a  mixed  number: 

II©.  Divide  the  numerator  by  the  denominator. 
If  there  is  a  remainder  write  it  above  the  denominator 
for  the  fractional  part.  Thus: 

33.  How  many  thirds  in  25  thirds? 

SOLUTION. 

25-^3:=  AT-  or  8i,  the  answer. 

<_>  o  * 

34.  How  many  fifths  in  18  fifths?  22  fifths?  25 
fifths?  30  fifths?  32  fifths?  40  fifths?  38  fifths? 

35.  How  many  ninths  in  24  ninths  ?  36  ?  84  ?  97  ? 
43?  96?  43?  82?  46?  22?  28?  64?  73? 

36.  How  many  boxes  candles  in  ^  ?   ££ft 

2 Ji  9   JL4  9  Y  9    2JI  9   !?.  9   A|  9   %L  9 


LESSON   YI. 

37.  If  one  pound  of  butter  costs  3-J-  cents,  what  is 
3  pounds  worth  ? 

Ans.  3  times  as  much  as  1  pound,  or  3  times 
3    cents. 


QUESTIONS   INVOLVING  F3ACTIOSS.  '  101 

SOLUTION. 

3  pounds  at  3  cents  are  9  cents.  3  pounds  at  \ 
cent  are  |  cents  or  1£  cents.  9  cents  and  1£  cents= 
10£  cents.  Or,  better,  reduce  the  mixed  numbers  to 
an  improper  fraction.  Multiply  the  numerator  by 
the  given  number  and  divide  the  product  by  the  de- 
nominator. 

Thus:  3^ r  cents = |  cents.  J  cents  x  3  =3f  cents 
or  10|,  the  answer. 

38.  If  one  hat  cost  3£  dollars  what  is  the  value  of 
4  hats  ?  5  hats  ?  G  hats  ?  7  hats  ?  8  hats  ?   9  hats  ? 
10  hats? 

39.  If  1  cap  cost  2^  dollars,  what  is  the  value  of 
2  caps  ?  3  caps  ?  4  caps  ?  5  caps  ?  6  caps  ? 

40.  If  1  pound  of  butter  is  worth  12  J  cents,  what  is 
the  value  of  20  pounds  ?  22  pounds  ?  18  pounds  ?  36 
pounds  ?  3  pounds  ?  G  pounds  ?  4  pounds  ? 


LESSOR  VII. 

41.  If  2|  boxes  of  cheese  are  worth  14  dollars, 
how  much  is  1  box  worth  ? 

If  2f  boxes  cheese  are  worth  14  dollars,  1  box 
would  be  worth  as  many  doll'ars  as  2f  are  contained 
in  14  dollars. 

1st  reduce  2|-  to  an  improper  fraction,  2-|— §. 

To  divide  by  a  fraction,  invert  the  divisor,  that  is, 
make  the  numerator  take  the  place  of  the  denomina- 
tor, and  the  denominator  the  place  of  the  numerator. 
9* 


1Q2  INTERMEDIATE   AEITHMETIC. 

Multiply  the  numerator  thus  found  by  the  given  num- 
ber, and  divide  the  result  by  the  new  denominator. 


SOLTJTIOX. 


14-j-|=14  xf=-=-4/.     424-8=5f,  the  answer. 

42.  I  gave  22  dollars  for  12  yards  and  %  of  cloth. 
What  was  the  cost  per  yard  ?  Ans.  $1.76. 

43.  Divide  GO  by  2£.  Ans.  24. 

44.  Divide  30  by  3£.  Ans.  9^. 
45    Divide  46  by  4£.                                 Ans.  1I£S. 

46.  Divide  89  by  2£.  Ans.  41-J-4J-. 

47.  Divide  73  by  9-£.  ^Lws.  7f|. 

48.  Divide  24  by  5f.  -4ws.  4f$. 

49.  Divide  92  by  6f  .4/w.  14f|. 

50.  Divide  100  by  8TV-  -4W5.  12|f. 

51.  Divide  200  by  12  J.  Ans.  16. 

52.  Divide  400  by  13f  Ans.  30f|-. 

53.  Divide  288  by  8TV  Ans.  32|f. 

54.  Divide  673  by  9TV  Ans.  68ff. 

55.  Divide  941  by  8i  Ans.  11  Off. 

56.  Divide  896  by  8f.  Ans.  102|, 

57.  A  mile  is  5280  feet  long.     A  rod  is  16£  feet. 
How  many  rods  in  a  mile  ?  Ans.  320. 

58.  Divide  2264  rods  by  5|  rods.         Ans.  411T\. 

59.  A  mile  is  1760  yards  long.     5|  yards  make  a 
rod.     How  many  rods  in  1  mile  ?  Ans.  320. 

60.  I  paid  250  dollars  for  20-f  yards  silk.    How 
much  is  that  per  yard?  Ans.  $5.22  J. 


CONTRACTIONS.  103 

61.  I  gave  22  cents  for  13^  crackers.     How  much 
is  1  cracker  worth?  Ans.  Itf. 

62.  I  sold  22^  pounds  cheese  for  215  cents.     At 
what  price  per  pound  was  it  sold  ?  Ans.  9J-. 


CONTRACTIONS. 
LESSON  VIII. 

111.  CONTRACTIONS  are  short  methods  of  per- 
forming multiplication  or  division. 

CASE  I. 

113.  To  multiply  any  number  by  25. 

EXAMPLE. 

Multiply  246  by  25. 

25  is  one-fourth  of  100.     To  mul-  OPERATION. 

tiply  any  number  by  100,  annex  two  4^24600 

O's.  By  multiplying  246  by  100,  we 
increase  its  value  4  times.  By  di- 
viding by  4,  we  get  the  true  product.  Ans.  6150. 

From  which  we  derive  the  following 
KULE. 


3.  Annex  two  O's  to  the  multiplicand,  and  di- 
vide by  4. 


104 


INTERMEDIATE   ARITHMETIC. 


1.  Multiply 

2.  Multiply 

3.  Multiply 

4.  Multiply 

5.  Multiply 

6.  Multiply 

7.  Multiply 

8.  Multiply 

9.  Multiply 
10.  Multiply 


EXAMPLES. 

1234  by  25. 
12345  by  25. 
2345  by  25. 
23456  by  25. 
3456  by  25. 
34567  by  25. 
4567  by  25. 
45678  by  25. 
5678  by  25. 
56789  by  25. 


Ans.  30850. 
Ans.  308625. 

Ans.  58625. 
Ans.  586400. 

Ans.  86400. 

Ans.  864175. 

Ans.  114175. 

Ans.  1141950. 

Ans.  141950. 

Ans.  1419725. 


LESSON  IX. 

CASE  II. 

114.  To  multiply  by  12$. 

EXAMPLES. 

1.  Multiply  126  by  '12$. 

Annex  two  ciphers  to  the  multi- 
plicand (126),  and  divide  the  number 
thus  formed  by  8,  because  12i-  is  the 
eighth  part  of  100.  Ans.  1575. 

2.  Multiply  1234  by  12$-. 

3.  Multiply  12340  by  12$. 

4.  Multiply  2340  by  12*. 

5.  Multiply  3456  by  12|. 

6.  Multiply  34560  by  12$. 

7.  Multiply  345600  by  13$. 


OPEBATION. 

8)12600 
1575 

Ans.  15425. 

Ans.  154250. 

Ans.  29250. 

Ans.  42200. 

Ans.  432000. 

Ans.  4320000. 


CONTRACTIONS.  105 

8.  Multiply  386400  by  12|.  A:is.  4830000. 

9.  Multiply  246800  by  12$.  Ans.  3085000. 

10.  Multiply  3698700  by  12£  Ans.  46233750. 

11.  Multiply  692460  by  12£.  Ans.  8655750. 


LESSON  X. 

CASE   III. 

115.  To  multiply  by  33 J. 
331-  is  J  of  100.     To  multiply  by  33£ : 
Annex  two  O's  to  the  multiplicand  and  divide  by  3. 
EXAMPLES. 

1.  Multiply  12345  by  33J. 

OPERATIOy. 

3)1234500 

411500,  the  answer. 

2.  Multiply  123456  by  33J.  Ans.  4115200. 

3.  Multiply  23456  by  33J.  Ans.  781866|. 

4.  Multiply  70368  by  33£.  Ans.  2345600. 

5.  Multiply  211104  by  33^.  Ans.  7036800. 

6.  Multiply  633312  by  331.  Ans.  21110400. 

7.  Multiply  1899936  by  33-J.  Ans.  63331200. 

8.  Multiply  5699808  by  33|.  Ans.  189993600. 

9.  Multiply  17099424  by  33  J.  Ans.  569980800. 

10.  Multiply  51298272  by  33J.   Ans.  1709942400. 

11.  Multiply  153894816  by  33^.   Ans.  5129827200. 


106 


INTERMEDIATE   AEITHMETIC. 


LESSON  XT. 
116.  To  multiply  by  125. 

125  is  i  of  1000.     To  multiply  by  1000,  annex  3 
ciphers  to  the  multiplicand.     To  multiply  by  125  : 

Annex  three  O's  to  the  multiplicand  and  divide  by  8. 

EXAMPLES. 

1.  Multiply  12344  by  125. 

OPERATION. 

8)12344000 


2.  Multiply 

3.  Multiply 

4.  Multiply 

5.  Multiply 

6.  Multiply 

7.  Multiply 

8.  Multiply 
9    Multiply 

10.  Multiply 

11.  Multiply 


1543000 

98752  by  125. 
790016  by  125. 
6320128  by  125. 
50460984  by  125. 
888888  by  125. 
7111104  by  125. 
56888832  by  125. 
404040  by  125. 
3232320  by  125. 
25858560  by  125. 


Ans.  12344000. 

Ans.  98752000. 

Ans.  790016000. 

Ans.  6307623000. 

Ans.  111111000. 

Ans.  888888000. 

Ans.  7111104000. 

Ans.  50505000. 

Ans.  404040000. 

Ans.  3232320000. 


LESSON  XII. 
117.  To  multiply  by  any  number  of  9's. 

By  adding  1  to  any  number  consisting  wholly  of 
9's,  we  obtain  a  number  the  first  figure  of  which  is 


CONTRACTIONS.  107 

1,  followed  by  as  many  O's  as  there  are  9?s  in  the 
first  number.  Therefore,  to  multiply  by  any  number 
of  9's,  add  as  many  O's  to  the  multiplicand  as  there 
are  9's  in  the  multiplier.  This  will  give  a  number 
greater  than  the  required  product  by  1  times  the 
original  multiplicand.  Subtract  the  first  multipli- 
cand from  the  new  multiplicand,  and  the  difference 
will  be  the  true  result. 

EXAMPLES. 

1.  Multiply  12345  by  99.  Ans.  1222155. 

OPERATION. 

1234500     Minuend. 
12345     Subtrahend. 


1222155     Remainder. 

Ans.  1222155. 

2.  Multiply  12  by  9.  Ans.  108. 

3.  Multiply  12  by  99.  Ans.  1188. 

4.  Multiply  123  by  9.  Ans.  1107. 

5.  Multiply  123  by  99.  Ans.  12177. 

6.  Multiply  1234  by  99.  Ans.  122166. 

7.  Multiply  1234  by  999.  Ans.  1232766. 

8.  Multiply  12345  by  999.  Ans.  12332655. 

9.  Multiply  1.2345  by  9999.  Ans.  123437655. 

10.  Multiply  123456  by  9999.     Ans.  1234436544. 

11.  Multiply  123456  by  99999,  Ans.  12345476544. 

12.  Multiply  123456  by  999.          Ans.  123332544. 

13.  Multiply  1234567  by  9999.  Ans.  12344435433. 

14.  Multiply  23456  by  99999.     Ans.  2345576544. 


108  INTERMEDIATE   ARITHMETIC. 

15.  Multiply  34567  by  999.  Ans.  34532433. 

16.  Multiply  87643  by  9999.         Ans.  876342357. 

17.  Multiply  76422  by  9999.         Ans.  764143578. 

18.  Multiply  28764  by  99999.     Ans.  2876371236. 

19.  Multiply  876492  by  99999.  Ans.  87648323508. 

20.  Multiply  332233  by  99999.  Ans.  33222967767, 

21.  Multiply  8764321  by  99999. 

Ans.  876423335679. 

22.  Multiply  123456  by  999999. 

Ans.  123455876544. 

23.  Multiply  2345678  by  999999. 

Ans. 

24.  Multiply  3456789  by  99999999. 

Ans. 

25.  Multiply  123456789  by  99999999. 

Ans. 

26.  Multiply  213141516171  by  99999999. 

Ans. 

27.  Multiply  31451681923  by  99999999. 

Ans. 


LESSON  XIII. 

CONTRACTIONS  IN   DIVISION. 

118.  To  divide  by  25. 

If  we  multiply  the  dividend  by  4,  we  get  a  product 
4  times  too  great.  To  find  the  true  quotient,  AVC 
must  divide  by  a  number  4  times  greater  than  the 
given  divisor.  To  divide  by  25,  multiply  the  divi- 


CONTRACTIONS. 


109 


dend  by  4  and  divide  the  product  by  100,  which  is 
the  same  as  cutting  off  two  figures  from  the  right  of 
the  result. 

EXAMPLES. 


I.  Divide  1234  by  25. 

Multiply  by  4,  and  from  the  prod- 
uct cut  off  2  figures.  The  answer  is 
49  times  and  -ffc  remaining. 

2.  Divide  12345  by  25. 

3.  Divide  23456  by  25. 

4.  Divide  34567  by  25. 

5.  Divide  45678  by  25. 

6.  Divide  56789  by  25. 

7.  Divide  678910  by  25. 

8.  Divide  78910  by  25. 

9.  Divide  891011  by  25. 
10.  Divide  910100  by  25. 

II.  Divide  12000  by  25. 

12.  Divide  23000  by  25. 

13.  Divide  23400  by  25. 

14.  Divide  345000  by  25. 

15.  Divide  456000  by  25. 


OPERATION. 

1234 
4 

49)36 

Ans.  493T*0V 
Ans.  93  8  y2^. 
Ans.  1382TV^. 
Ans.  1827-^. 
Ans.  2271-^0-. 
Ans.  2  71 5GTV0-. 
Ans.  3156-JvV. 
Ans.  35640^. 
Ans.  36404. 
Ans.  480. 
Ans.  920. 
Ans.  936. 
Ans.  1380. 
Ans.  18240. 


LESSON  XIV. 
119.  To  divide  by  33^. 

If  we  multiply  the  dividend  by  3,  we  get  a  num- 
ber three  times  too  great.     To  find  the  true  quotient, 
we  must  divide  by  a  number  three  times  greater 
than  the  given  divisor. 
10 


110 


INTERMEDIATE   ARITHMETIC. 


1 2O.  331-  is  ^  Of  100.  Multiplying  by  3  and 
dividing  by  100  is  the  same  as  dividing  by  33-|-.  To 
divide  by  33^,  multiply  the  dividend  by  3  and 
divide  the  product  by  100,  by  taking  off  two  figures 
from  the  right. 


EXAMPLES. 

I.  Divide  2468  by  33-J-. 

1st.  Multiply  the  dividend  by  3. 
2d.  From  the  product  strike  off  two 
figures.  The  true  answer  is  74  and 

remaining. 

2.  Divide  12345  by  33 J. 

3.  Divide  23456  by  33J. 

4.  Divide  34567  by  33|. 

5.  Divide  45678  by  33f 

6.  Divide  56789  by  33 J. 

7.  Divide  678910  by  33J. 

8.  Divide  78912  by  3 3|. 

9.  Divide  89123  by  33J. 
10.  Divide  91234  by  33J. 

II.  Divide  43219  by  33J. 

12.  Divide  32189  by  33|. 

13.  Divide  198756  by  33J. 

14.  Divide  246890  by  33f 


OPERATION. 

2468 
3 


7404 


Ans. 

Ans. 
Ans. 
Ans. 
Ans.  1703T<W 

Am.  20367TW 
Ans. 
Ans. 

Ans.  2737^-0-. 
Ans. 
Ans. 
Ans. 


Ans. 


CONTEACTIOXS. 


LESSON  XV. 


Ill 


121.  To  divide  by  12$: 

Multiply  the  dividend  by  8  and  divide  the  product 
ly  100  (because  12$  is  \  of  100). 


EXAMPLES. 

1.  Divide  86432  by  12$.                           OPEEATION. 
1st.  Multiply  by  8.     2d.  Cut  off             86432 
two  figures  from  the  right  of  the  re«                     R 

suit.     The  answer  is  6914  and  -f£ 

T  ^*®~                _ 

maining. 

6914|56 

2.  Divide 

987  by  12$. 

Ans.  78  ,'„«,. 

3.  Divide 

8976  by  12$. 

Ans.  nSyJhp 

4.  Divide 

6798  by  12$. 

Ans.  543^/g-. 

5.  Divide 

7968  by  12$. 

Ans.  637^0- 

6.  Divide 

4312  by  12$. 

Ans.  344^. 

7.  Divide 

4132  by  12$. 

Ans.  320^^-. 

8.  Divide 

4321  by  12$. 

Ans.  345-^-. 

9.  Divide 

76425  by  12$. 

Ans.  6114. 

10.  Divide 

46725  by  12$. 

Ans.  5738. 

11.  Divide 

67452  by  12$. 

Ans.  5396^. 

12.  Divide 

47625  by  12$. 

Ans.  3810. 

13.  Divide 

18793  by  12$. 

Ans.  1503^(5*0. 

14.  Divide 

78139  by  12$. 

Ans.  6251^%. 

15.  Divide 

17893  by  12$. 

Ans.  1431-^j-. 

16.  Divide 

38719  by  12$. 

Ans.  3097T5o?o-. 

17.  Divide 

248763  by  12$. 

Ans.  19901^. 

18.  Divide 

427836  by  12$. 

Ans.  34226^. 

19.  Divide 

742886  by  12$. 

Ans.  59430T8<yV 

20.  Divide 

3217546  by  12$. 

Ans.  257403^. 

112 


INTERMEDIATE  ARITHMETIC. 


LESSON  XVI. 
1SS.  To  divide  by  125. 
125  is  £  of  1000. 

Multiply  the  dividend  by  8  and  divide  the  product 
by  1000,  by  taking  three  figures' from  the  right  of  the 
result. 


EXAMPLES. 


1.  Divide  186254  by  125. 

1st.  Multiply  186254  by  8.  2d. 
From  the  result,  1490032  cut  off  three 
figures.  The  answer  is  1490r§f-g-. 


OPEEATION. 

186254 

8 

1490|032 


2.  Divide  12345  by  125. 

3.  Divide  23456  by  125. 

4.  Divide  34567  by  125. 

5.  Divide  45678  by  125. 

6.  Divide  56789  by  125. 

7.  Divide  678910  by  125. 

8.  Divide  7891011  by  125. 

9.  Divide  891234  by  125. 

10.  Divide  9123456  by  125. 

11.  Divide  1111111  by  125. 

12.  Divide  222222  by  125. 

13.  Divide  3333333  by  125. 

14.  Divide  6666666  by  125. 

15.  Divide  46644664  by  125. 

16.  Divide  8282828282  by  125. 


Am. 
Am. 
AM. 
Am. 

Am.  454-^VV 
Am.  5431TVV>0-. 

Am.  63128-^. 
Am.  7129iV&. 

Ans.  72987T%Vi>  • 
Ans.  8888TVVV 
Am.  1777TVoV 

Ans.  26666Tyv4o- 

Am.  52222T2o2A- 
Am.  373157fVft- 


MISCELLANEOUS   EXAMPLES  113 

17.  Divide  544554455445  by  125. 

Am.  4356435643^^. 

18.  Divide  38765433456783  by  125. 

Am.  310123467654  jV./u- 

19.  Divide  12345677654321  by  125. 

Am.  987652212363^%. 


EXAMPLES  INVOLVING  THE  FOREGOING  RULES. 
LESSON  XVI 

123.  1.  I  bought  20  hogsheads  sugar  @  $30 
per  hhd.,  and  sold  them  for  $45  per  hhd.  What  did 
I  gain?  Am.  $300. 

2.  I  bought  25  hhds.  sugar,  250  pounds  each,  @ 
12£  cents  per  pound,  and  sold  20  hhds.  of  it  for  15 
cents.     How  much  did  I  pay  for  the  sugar  ?     How 
much  did  I  receive  for  what  I  sold?    How  many 
pounds  of  sugar  had  I  left,  and  what  was  it  worth, 
@  20  cents  per  pound.  First  Ans.  $781.25. 

3.  A  grocer  bought  14  hogsheads  rice,  200  pounds 
each,  @  33J  cents  per  pound,  and  sold  it  for  34  cents. 
How  much  did  he  gain  ?  Ans.  $18.66f. 

4.  Bought  27  oxen  @  40  dollars,  32  cows  @  25 
dollars,  and  gave  in  payment  25  horses,  valued  @  75 
dollars  each.     How  much  do  I  owe  ?  Ans.  $5. 

5.  The  distance  between  two  cities  is  840  miles. 

10* 


114  INTERMEDIATE  ARITHMETIC. 

If  I  can  travel  42  miles  a  day  on  foot,  in  how  many 
days  will  I  accomplish  the  journey?      Am.  20  days. 

6.  I  sold  S.  Brown  200  kegs  nails,  @  3  dollars ; 
100  thousand  feet  lumber,  @  12£  dollars  per  thou- 
sand.    How  much  does  he  owe  me  ?  Am.  $725. 

7.  In  one  year  there  are  365  days.     How  many 
days  in  12  years  ?  Am.  4380  days. 

8.  Bought  of  Jas.  Philips  200  acres  of  land,  @  $25 
per  acre,  and  gave  him  in  payment  25  horses,  valued 
at  $200  each.     How  mucli  does  he  owe  me  ? 

Am.  Nothing. 

9.  Make  out  the  following  bill :  50  barrels  flour, 
@  $5;  40  hundredweight  cheese,  @   8  dollars  per 
hundredweight ;  15  boxes  salmon,  @  7  dollars  per 
box.     How  much  did  the  bill  amount  to  ?  Am.  $675. 

10.  I  bought  2  casks  oil,  184  gallons  each,  @  $1 
per  gallon.     22  gallons  having  leaked  out,  I  sell  the 
remainder  for  105  cents  a  gallon.     Do  I  gain  or  lose, 
and  how  much?  Ans.  Lost  $27.80. 

11.  I  bought  240  acres  of  land,  @  $10  an  acre, 
and  sold  120  acres  of  it  to  Wm.  Brown,  @  $20  an 
acre.     How  much  is  the  balance  worth,  at  $13  an 
acre,  and  how  much  do  I  gain?  Am.  $1560. 

13.  I  gave  1728  dollars  for  a  certain  number  of 
barrels  of  flour,  valued  at  12  dollars  a  barrel.  I  sold 
the  flour  for  14  dollars  a  barrel,  How  many  barrels 
did  I  buy?  How  much  did  I  receive?  How  much 
profit  did  I  make  ? 

Ans.  Bought  144  bbls.;  profit,  $288. 


MISCELLANEOUS   EXAMPLES.  115 

LESSON  XVIII. 

14.  The  sum   of  two   numbers   is    123456.     The 
smaller  number  is  1234.     What  is  the  larger  num- 
ber? Am.  122222. 

15.  The  dividend   is    1864.      The   divisor   is   2. 
What  is  the  quotient  ?  Ans.  932. 

16.  The  quotient  is  932.    The  divisor  is  2.    What 
is  the  dividend  ?  Ans.  1864. 

17.  The  quotient  is  932.     The  dividend  is  1864. 
What  is  the  divisor.  Ans.  2. 

18.  I  have  a  farm  of  21000  acres,  valued  at  2  dol- 
lars per  acre.     I  bequeath  £  of  it  to  my  son,  Robert ; 
£  of  the  remainder  to  my  daughter,  Jane ;  and  the 
remainder  I  reserve  for  myself.     How  much  is  the 
v/hole  farm  worth  ?     How  many  acres  did  Jane  re- 
ceive for  her  share,  and  what  is  her  share  worth,  at 

5  dollars  per  acre?     How  many  acres  did  my  son 
Robert  receive,  and  what  is  the  value  of  his  share,  at 

6  dollars  an  acre  ?     How  many  acres  did  I  reserve^ 
and  what  is  the  value  of  my  portion,  at  12  £  dollars 
an  acre?     What  is  the  value  of  the  whole  farm  as 
divided  above  ?  Ans. 

19.  I  buy  molasses  to  the  amount  of  1800  dollars, 
at  the  rate  of  4  dollars  per  barrel.     How  many  bar- 
rels did  I  buy  ?  Ans.  450. 

20.  A  train  of  twelve  cars  brought  down"  cotton 
on  the  Jackson  Railroad.     Each  car  contained  20 
bales,  and  every  bale  weighed  420  pounds.     What 
was  the  value  of  the  cotton,  at  12^  cents  per  pound  ? 

Ans.  $12600. 


116 


INTERMEDIATE   ARITHMETIC. 


21.  The  distance  from  the  earth  to  the  moon  is 
two  hundred  and  forty  thousand  miles.     How  long 
would  it  take  a  steam-car  to  get  there,  at  the  rate  of 
100  miles  per  hour,  allowing  24  hours  to  the  day? 

Ans.  100  days. 

22.  The  population  of  Pekin  is  estimated  at  two 
millions.     If  each  person  in  Pekin  should  eat  one 
and  a  half  pound  of  rice  per  day,  how  many  casks 
of  rice  would  be  consumed,  allowing  380  Ibs.  to  a 
cask  ?    What  would  be  the  value  of  the  rice,  @ 
cents  a  pound  ?  First  Ans.  3000009. 

23.  London  has  two  millions  eight  hundred  thou- 
sand  inhabitants.     How   many   pounds   of  mutton 
would  be  consumed,  if  every  inhabitant  consumed  If 
pound   per   day  ?      How   much    per   week  ?     How 
many  sheep  would  be  required  to  supply  London  per 
day,  if  a  sheep  weighed  65  pounds  ?  per  week  ? 

First  Ans.  4900000. 

24.  I  purchased  23640  gallons  of  molasses.    How 
many  barrels  did  I  buy,  if  every  barrel  contained  40 
gallons  ?     If  I  sold  the  molasses  for  20  dollars  a  bar- 
rel, bow  much  would  I  have  to  receive  ?    If  I  sold 
the  molasses  at  55  cents  a  gallon,  how  much  would  I 
receive,  and  what  would  my  profit  be,  if  I  bought 
the  molasses  at  45  cents  a  gallon  ? 

First  Ans.  591  barrels. 

25.  The  building  of  a  bridge  cost  as  follows :  25 
carpenters,  18  days,  @  3  dollars  per  day;  30  brick- 
layers, 18  days,  @  3  dollars  per  day ;  20  laborers,  1! 
days,  @  2  dollars  per  day;  10000  bricks,  @  $13  per 
thousand;  64000  feet  lumber,  @  18  dollars  perthou- 


MISCELLANEOUS   EXAMPLES.  117 

sand ;  2  plans  of  the  bridge,  @  25  dollars  each.  I 
received  30000  dollars  for  making  the  bridge.  Do  I 
gain  or  lose,  and  how  much  ?  What  would  my  sal- 
ary as  contractor  be,  if  I  was  20  days  superintending 
the  work  ?  Am. 

26.  Horrel,  Gayle  &  Co.,  bought  ofJas.  Brown: 
200  sacks  of  corn,  @  3  dollars  per  sack;  100  sacks  of 
bran,  @  2  dollars  per  sack;  25  barrels  of  pork,  @  28 
dollars  per  barrel;  30  barrels  of  beef,  @  21  dollars 
per  barrel.     What  is  the  amount  of  the  bill  ? 

Am.  $2130. 

LESSON  XIX. 

27.  If  there  are  365  days,  of  24  hours  each,  in  one 
year,  how  many  hours  in  ten  years?         Ans.  87600. 

28.  If  there  are  5280  feet,  of  12  inches  each,  in  1 
mile,  how  many  inches  in  10  miles?        Am.  633600. 

29.  There  320  boys  in  the  Bienville  school.     If 
each  boy  would  contribute  10  cents  for  charitable 
purposes,  how  much  would  each  beggar  receive,  if 
25  made  application  for  his  share  ?  Ans.  $1.28. 

30.  In  one  mile  there  are  63360  inches.     A  brick 
is  8  inches  long.     How  many  bricks  would  make  1 
mile  ?  2  miles  ?  Last  Ans.  15840. 

31.  My  school  books  cost  as  follows:  1  reader, 
135  cents;  1   speller,  45  cents;  1  slate,  20  cents ;  1 
arithmetic,   100  cents;    1  Scholar's  Companion,  95 
cents;  1  geography,  150  cents;  1  history,  200  cents. 
What  did  all  of  them  cost?  Ans.  $7.95. 

32.  If  12  inches  make  1  foot,  how  many  feet  in 
63360  inches?  Ans.  5280. 


118 


INTERMEDIATE    ARITHMETIC. 


33.  If  3  feet  make  1  yard,  how  many  yards  in  5280 
feet?  Ans.  1760. 

34.  Make  the  following  bill : 

2  Grammars,  @  55  cents      ,     .     .     . 

4  Histories  U.  S.,  @  150  cents .     .     . 

5  Scholar's  Companions,  at  80  cents  . 
8  Arithmetics,  @  70  cents    .... 

Total   . 816.70 

35.  If  1  grammar  costs  55  cents,  how  many  gram- 
mars can  I  buy  for  5555  cents  ?  Ans.  101. 

36.  What  cost  250  boxes  chalk,  at    12^  cents  per 
box?  Ans.  $31.25. 

37.  What  cost  250  barrels  beef,  at  25  dollars  per 
barrel?  Ans.  86250. 

38.  Multiply  2468  by  999,  by  contraction. 

Ans.  2465532. 

40.  If  a  ship  costs  40000  dollars,  what  will  ^  of  a 
ship  cost  ?     What  is  f  worth  ? 

Last  Ans.  25000  dollars. 

41.  20  men  work  70  days,  at  125  cents  per  day. 
How  much  will  their  wages  amount  to  ?    Ans.  $175. 

42.  Bought  4267  slates,  at  25  cents  apiece.     How 
much  did  they  cost?  Ans.  $1066.75. 

43.  If  1  hhd.  sugar  costs  200  dollars,  what  is  f  of 
it  worth?  Ans.  150  dollars. 

44.  If  a  steamboat  costs  42-000  dollars,  how  much 
is  f  of  it  worth  ?  Ans.  18000. 

45.  If  a   school-house   costs  12000  dollars,  how 
much  is  -f  of  it  worth  ?  Ans.  $9600. 


MISCELLANEOUS   EXAMPLES.  119 


LESSON  XX. 

46.  If  in  1   mile  there  are  5280  feet,  how  many 
miles  in  158400  feet?  Ans.  30  miles. 


.    BY   CONTRACTION. 

47.  I  bought  2460  Ibs.  butter,  at  12£  cents.     Re- 
quired the  amount  of  my  bill.  Ans.  $307.50. 

48.  I  sold  98  dozen  of  hats,  at  12£  dollars  a  dozen. 
How  much  was  my  bill  ?  Ans.  $1225. 

49.  To  25  men  I  gave  43  dollars  apiece.     How 
much  did  I  give  to  all  of  them  ?  Ans.  $1075. 

50.  John  S  my  the  bought   46   green   mules,   for 
which  he  paid  125  dollars  apiece.     How  much  did  he 
have  to  spend?  Ans.  85750. 

51.  James  Brown  bought  64  quires  of  paper,  at 
331  cents  a  quire.     How  much  was  the  amount  of 
the  bill?  Ans.  $21.33£. 

52.  If  1  hat  costs  8  dollars,  how  many  hats  can  I 
buy  for  640  dollars  ?  Ans.  80. 

53.  How  much  will  200000  feet  of  lumber  cost,  at 
12i-  dollars  per  thousand?  Ans.  $5000. 

54.  How  much  will  264000  bricks  cost,  at  12£ 
dollars  per  thousand  ?  Ans.  $3300. 

55.  I  received  24  loads  of  brick,  each  load  con- 
taining 500  bricks.     How  much  are  they  worth,  at 
33i  dollars  per  thousand  ?  Ans.  8400. 


120  INTERMEDIATE    ARITHMETIC. 

56.  How  much  will  the  following  bill  amount  to : 
24  thousand  bricks,  @  12^  dollars   per  thousand;  18 
thousand  feet  lumber,  @  25  dollars  per  thousand  ? 

Ans.  $750. 

57.  In   12  days  of  24  hours  each,  every  hour  60 
minutes,  every  minute  60  seconds,  how  many  sec- 
onds ?  Ans.  1036800. 

58.  In  25  carts  there  are  50  planks,  and  each 
plank  measures  14  feet.     How  much  is  the  lumber 
worth,  at  12£  cents  per  foot  ?  Ans.  $2187.50. 

59.  If  I  buy  lumber  at  12^  cents  per  foot,  and  sell 
at  125  dollars  per  thousand,  do  I  gain  or  lose,  and 
how  much  ?    Why  ?  Ans. 

60.  If  I  buy  a  calf  weighing  100  pounds,  at  12^ 
cents  a  pound,  and  sell  it  for  12 J  dollars,  how  much 
do  I  gain  or  lose  ?     Why  ?  Ans. 

61.  I  require  240000  feet  of  lumber  and  300000 
bricks.     John  Brown  says  that  he  will  sell  me  lum- 
ber at  25   dollars  per  thousand,  and  bricks  at   12| 
dollars  per  thousand.     Robert  Smith  says  that  he 
can  sell  cheaper  than  Brown.     He  says  that  I  can 
have  his  lumber  for  2|  cents  per  foot,  and  his  bricks 
for  l£  cents  apiece.     How  much  do  I  save  by  buying 
from.  Smith,  and  why  ?  Ans. 


UNITED   STATES   MONEY.  121 

CHAPTER    VII. 

PEACTICE  IN  UNITED  STATES  MONEY. 
LESSON  I. 


UNITED  STATES  MONEY  was  established 
by  the  Congress  of  the  United  States  in  1796  as  the 
legal  currency  of  the  United  States. 

In  1862,  Congress  issued  bills  of  credit  called 
United  States  Treasury  notes,  or,  as  they  are  com- 
monly called,  "  Greenbacks" 

126.  The  units  increase  ten  times  for  every 
figure  from  right  to  left,  and  decrease  in  the  same 
proportion  from  left  to  right. 

IS1?'.  The  dollars  are  separated  from  the  cents 
by  a  period  (  .  )  called  the  decimal  point.  All  the 
figures  to  the  left  of  the  period  are  dollars.  The  first 
figure  after  the  period  represents  dimes,  the  second, 
cents,  and  the  third,  mills.  Thus,  $18.625  is  read,  18 
dollars  6  dimes  (or  60  cents)  2  cents  and  5  mills  ;  or, 
18  dollars  62  cents  and  5  mills. 

128.  Since  it  takes  10  cents  to  make  1  dime? 
when  the  cents  are  less  than  10,  a  cipher  must  be 
placed  after  the  period  and  before  the  figure  repre- 
senting cents. 

EXAMPLE. 

Write  1  dollar  and  5  cents.  Ans.  $1.05. 

11 


122 


INTERMEDIATE    ARITHMETIC. 


129.  The  gold  coins  of  the  United  States  are 
the  double  eagle,  or  $20  gold  piece ;  the  eagle,  or  $10 
gold  piece ;  the  half  eagle,  or  $5  gold  piece ;  the 
quarter  eagle,  or  $2.50  gold  piece;  the  $3  gold  piece; 
and  the  $1  gold  piece. 

1 3O.  The  silver  pieces  are  the  dollar,  half  dollar, 
quarter  dollar,  dime,  half  dime  or  picayune,  and  3- 
cent  piece.     There  is  also  the  1-cent  piece,  made  of 
copper.     Nickels,  or  5-cent  pieces,  are  now  made  of 
nickel  and  copper. 

131.  This  sign,  8,  is  the  sign  of  dollars. 

TABLE    OF   UNITED   STATES   MONET. 


10  mills  make  1 

10  cents         "  1 

10  dimes        "  1 

10  dollars       "  1 


cent,  marked  ct. 

dime,  "  d. 

dollar,         "  $. 

eagle,          "  E. 


KEDUCTION  I. 

132.  To  reduce  dollars  to  dimes. 

RULE. 
Annex  1  cipher. 

EXAMPLES. 

1.  Reduce  $25  to  dimes. 

OPERATION". 

$25  X  10  —  250  dimes. 


Since  it  takes  10  dimes  to  make  1  dollar,  there 
will  be  10  times  as  many  dimes  as  dollars.  To  mul- 
tiply by  10,  annex  1  cipher  to  the  multiplicand. 


UNITED    STATES   MONEY.  123 

2.  How  many  climes  in  82?  Ans.  20  dimes. 

3.  How  many  dimes  in  $3  ?  Ans.  30  dimes. 

4.  How  many  dimes  in  84  ?  Ans.  40  dimes. 

5.  How  many  dimes  in  §15?  Ans.  150  dimes. 
G.  How  many  dimes  in  820?  Ans.  200  dimes. 

7.  How  many  dimes  in  $200  ?  Ans.  2000  dimes. 

8.  How  many  dimes  in  $250?  Ans.  2500  dimes. 

9.  How  many  dimes  in  $300  ?  Ans.  3000  dimes. 

10.  How  many  dimes  in  8208  ?  Ans.  2080  dimes. 

11.  How  many  dimes  in  8309?  Ans.  3090  dimes. 

12.  How  many  dimes  in  8390  ?  Ans.  3900  dimes. 

13.  How  many  dimes  in  848G?  Ans.  4860  dimes. 

14.  How  many  dimes  in  $846?  Ans.  84GO  dimes. 

15.  How  many  dimes  in  $24G  ?  Ans.  2460  dimes. 

16.  How  many  dimes  in  $642?  Ans.  6420  dimes. 

17.  How  many  dimes  in  $462  ?  Ans.  4620  dimes. 

18.  How  many  dimes  in  $8972  ? 

Ans.  89720  dimes. 


LESSON    II. 

DEDUCTION   II. 

133.  To  reduce  dollars  to  cents. 

RULE. 
Multiply  the  dollars  by  100. 

Because  it  takes  100  cents  to  make  a  dollar, 
therefore  there  will  be  100  times  as  many  cents  as 
there  are  dollars.  To  multiply  by  100,  annex  two 
ciphers  to  the  multiplicand. 


124  INTERMEDIATE    ARITHMETIC. 


EXAMPLES. 

1.  How  many  cents  in  25  dollars? 

Am.  2500  cents. 

OPERATION'. 

$25  X  100  =  2500  cents. 

2.  How  many  cents  in  24  dollars  ? 

Ans.  2400  cents. 

3.  How  many  cents  in  42  dollars  ? 

Ans.  4200  cents. 

4.  How  many  cents  in  58  dollars  ? 

Ans.  5800  cents. 

5.  How  many  cents  in  85  dollars  ? 

Ans.  8500  cents. 

6.  How  many  cents  in  976  dollars  ? 

Ans.  97600  cents. 

7.  How  many  cents  in  679  dollars? 

Ans.  67900  cents. 

8.  How  many  cents  in  796  dollars? 

Ans.  79600  cents. 

9.  How  many  cents  in  769  dollars? 

Ans.  76900  cents. 

10.  How  many  cents  in  287  dollars  ? 

Ans.  28700  cents. 

11.  How  many  cents  in  827  dollars? 

Ans.  82700  cents. 

12.  How  many  cents  in  872  dollars? 

Ans.  87200  cents. 

13.  How  many  cents  in  728  dollars  ? 

Ans.  72800  cents. 


UNITED    STATES    MOXKY.  125 

14.  How  many  cents  in  348  dollars  ? 

Ans.  34800  cents. 

15.  How  many  cents  in  976  dollars  ? 

Ans.  97600  cents. 
1C.  How  many  cents  in  834  dollars  ? 

Ans.  83400  cents. 

17.  How  many  cents  in  G79  dollars? 

Ans.  67900  cents. 

18.  How  many  cents  in  796  dollars? 

Ans.  79600  cents. 

19.  How  many  cents  in  643  dollars  ? 

Ans.  64300  cents. 

20.  How  many  cents  in  897  dollars? 

Ans.  89700  cents. 

21.  How  many  cents  in  241  dollars  ? 

Ans.  24100  cents. 

22.  How  many  cents  in  873  dollars  ? 

Ans.  87300  cents. 

23.  How  many  cents  in  946  dollars  ? 

Ans.  94600  cents. 

24.  How  many  cents  in  638  dollars  ? 

Ans.  63800  cents. 


LESSON   III. 

EEDUCTIOX  III. 

1 34.  To  reduce  dollars  to  mills. 

EULE. 

Annex  3  ciphers  to  the  dollars. 
11* 


126  INTERMEDIATE    ARITHMETIC. 

Since  it  takes  10  mills  to  make  1  cent,  and  100 
cents  to  make  1  dollar,  it  requires  10  x  100,  or  1000 
mills  to  make  1  dollar.  To  multiply  by  1000,  annex 
3  ciphers  to  the  multiplicand. 

EXAMPLES. 

1.  How  many  mills  in  4  dollars  ? 

OPERATION. 

4  X  1000  =  4000.     Ans. 

2.  How  many  mills  in  185  dollars  ?  Ans.  185000. 

3.  How  many  mills  in  851  dollars  ?  Ans.  851000. 

4.  How  many  mills  in  581  dollars?  Ans.  581000. 

5.  How  many  mills  in  518  dollars  ?  Ans.  518000. 

6.  How  many  mills  in  815  dollars  ?  Ans.  815000. 

7.  How  many  mills  in  723  dollars?  Ans.  723000. 

8.  How  many  mills  in  327  dollars  ?  Ans.  327000. 

9.  How  many  mills  in  237  dollars  ?  Ans.  237000. 

10.  How  many  mills  in  732  dollars?  Ans.  732000. 

11.  How  many  mills  in  914  dollars?  Ans.  914000. 

12.  How  many  mills  in  419  dollars?  Ans. 419000. 

13.  How  many  mills  in  149  dollars  ?  Ans.  149000. 

14.  How  many  mills  in  941  dollars?  Ans.  941000. 

15.  How  many  mills  in  186  dollars?  Ans.  186000. 

16.  How  many  mills  in  681  dollars?  Ans.  681000. 

17.  How  many  mills  in  618  dollars  ?  Ans.  618000. 

18.  How  many  mills  in  816  dollars?  Ans.  816000. 

19.  How  many  mills  in  861  dollars?  Ans.  861000. 

20.  How  many  mills  in  732  dollars  ?  Ans.  732000. 

21.  How  many  mills  in  723  dollars  ?  Ans.  723000. 

22.  How  many  mills  in  327  dollars?  Ans.  327000. 

23.  How  many  mills  hi  372  dollars?  Ans.  372000. 


UNITED   STATES  MONEY.  127 

LESSON    IV. 

DEDUCTION   IV. 

.  To  reduce  dimes  to  cents. 

RULE. 
Annex  1  cipher  to  the  multiplicand. 

Since  it  takes  10  cents  to  make  1  dime,  there  will 
be  1 0  times  as  many  cents  as  dimes.  To  multiply  by 
10,  annex  1  cipher. 

EXAMPLES. 

1.  Reduce  10  dimes  to  cents. 

OPEEATIOX. 

10  X  10  =  100  cents. 

2.  Reduce  4  dimes  to  cents.  Am.  40  cents. 

3.  Reduce  5  dimes  to  cents.  Ans.  50  cents. 

4.  Reduce  6  dimes  to  cents.  Ans.  60  cents. 

5.  Reduce  7  dimes  to  cents.  Ans.  70  cents. 

6.  Reduce  8  dimes  to  cents.  Ans.  80  cents. 

7.  Reduce  9  dimes  to  cents.  Ans.  90  cents. 

8.  Reduce  10  dimes  to  cents.  Ans.  100  cents. 

9.  Reduce  11  dimes  to  cents.  Ans.  110  cents. 
10.  Reduce  12  dimes  to  cents.  Ans.  120  cents. 


128 


I:NTEKMEDIATE  ARITHMETIC. 


LESSON   V. 

EEDUCTION   V. 

136.  To  reduce  cents  to  mills. 

EULE. 
Annex  1  cipher. 

Since  it  takes  10  mills  to  make  1  cent,  there  will 
be  10  times  as  many  mills  as  cents.  To  multiply  by 
10,  annex  1  cipher. 

EXAMPLES. 

1.  Reduce  5  cents  to  mills. 

OPERATION. 

5  x  10  =  50  mills. 

2.  Reduce  6  cents  to  mills.  Ans.  60  mills. 

3.  Reduce  7  cents  to  mills.  Ans.  70  mills. 

4.  Reduce  8  cents  to  mills.  Ans.  80  mills. 

5.  Reduce  9  cents  to  mills.  Am.  90  mills. 

6.  Reduce  10  cents  to  mills.  Ans.  100  mills. 


LESSON   VI. 

SEDUCTION  VI. 

1 37.  To  reduce  mills  to  cents. 

EULE. 
Divide  ~by  10. 

To  divide  by  10,  cut  off  1  figure  from  the  right. 


UNITED   STATES   MONEY.  129 

EXAMPLES. 

1.  Reduce  200  mills  to  cents. 

OPERATION. 

200  -f-  10  =  200  or  20  cents. 

2.  Reduce  340  mills  to  cents.  "      Am.  34  cents. 

3.  Reduce  430  mills  to  cents.  Am.  43  cents. 

4.  Reduce  640  mills  to  cents.  Ans.  64  cents. 

5.  Reduce  460  mills  to  cents.  Ans.  46  cents. 

6.  Reduce  280  mills  to  cents.  Ans.  28  cents. 
1.  Reduce  820  mills  to  cents.  Ans.  82  cents. 

8.  Reduce  490  mills  to  cents.          Ans.  49  cents. 

9.  Reduce  940  mills  to  cents.          Ans.  94  cents. 
10.  Reduce  62800  mills  to  cents.  Ans.  6280  cents. 


LESSON"   VII. 

REDUCTION   VII. 

138.  To  reduce  mills  to  dimes. 

ETJLE. 
Divide  by  100. 

To  divide  by  100,  cut  off  2  figures  from  the  right. 

EXAMPLES. 

1.  Reduce  2400  mills  to  dimes. 

OPERATION-. 

2400  -^  100  =  24  dimes. 

2.  Reduce  1800  mills  to  dimes.    Ans.  18  dimes. 


130  INTERMEDIATE   ARITHMETIC. 

3.  Reduce  2800  mills  to  dimes.     Ans.  28  dimes. 

4.  Reduce  1600  mills  to  dimes.     Ans.  16  dimes. 

5.  Reduce  18600  mills  to  dimes.  Ans.  186  dimes. 

6.  Reduce  98700  mills  to  dimes.  Ans.  987  dimes. 

7.  Reduce  27400  mills  to  dimes.  Ans.  274  dimes. 

8.  Reduce  87400  mills  to  dimes.  Ans.  874  digies. 


LESSON   VIII. 

KEDUCTIOX   VIII. 

139.  To  reduce  mills  to  dollars. 

KULE. 
Divide  ly  1000. 

To  divide  by  1000,  cut  off  3  figures  from  the 
right  of  the  number. 

1000  mills  make  1  dollar. 

EXAMPLES. 

1.  Reduce  24000  mills  to  dollars. 

OPERATION. 

24000  -f-  1000  =  24  dollars. 

2.  Reduce  64000  mills  to  dollars. 

Ans.  64  dollars. 

3.  Reduce  78000  mills  to  dollars. 

Ans.  78  dollars. 

4.  Reduce  96000  mills  to  dollars. 

Ans.  96  dollars. 

5.  Reduce  89000  mills  to  dollars. 

Ans.  89  dollars. 


UNITED    STATES   MONEY.  131 

C.  Reduce  746000  mills  to  dollars. 

Ans.  746  dollars. 

7.  Reduce  740000  mills  to  dollars. 

Ans.  740  dollars. 

8.  Reduce  8GOOOO  mills  to  dollars. 

Ans.  860  dollars. 

9.  Reduce  980000  mills  to  dollars. 

Ans.  980  dollars. 
10.  Reduce  8989000  mills  to  dollars. 

Ans.  8989  dollars. 

LESSON    IX. 

REDUCTION   IX. 

14:0.  To  reduce  cents  to  dimes. 

RULE. 
Divide  ly  10. 

To  divide  "by  10,  cut  off  1  figure  from  the  right. 
10  cents  make  1  dime. 

EXAMPLES. 

1.  Reduce  280  cents  to  dimes. 

OPERATION. 

280  -f-  10  =  28  dimes. 

2.  Reduce  760  cents  to  dimes.  Ans.  76  dimes. 

3.  Reduce  890  cents  to  dimes.  -  Ans.  89  dimes. 

4.  Reduce  980  cents  to  dimes.  Ans.  98  dimes. 

5.  Reduce  1760  cents  to  dimes.  Ans.  176  dimes. 

6.  Reduce  4300  cents  to  dimes.  Ans.  430  dimes. 

7.  Reduce  870  cents  to  dimes.  Ans.  87  dimes. 

8.  Reduce  960  cents  to  dimes.  Ans.  96  dimes. 


132  INTERMEDIATE  ARITHMETIC. 


LESSON    X. 

KEDTJCTION  X. 

141.  To  reduce  cents  to  dollars. 

RULE. 
Divide  by  100. 
To  divide  by  100,  cut  off  2  figures  from  the  right. 
100  cents  make  1  dollar. 

EXAMPLES. 

1.  Reduce  248000  cents  to  dollars. 

OPERATION. 

248000  -^  100  =  2480  dollars. 

2.  Reduce  18600  cents  to  dollars. 

Ans.  186  dollars. 

3.  Reduce  98700  cents  to  dollars. 

Ans.  987  dollars. 

4.  Reduce  87900  cents  to  dollars. 

Ans.  879  dollars. 

5.  Reduce  34200  cents  to  dollars. 

Ans.  342  dollars. 

6.  Reduce  672800  cents  to  dollars. 

Ans.  6728  dollars. 

7.  Reduce  246200  cents  to  dollars. 

Ans.  2462  dollars. 

8.  Reduce  297600  cents  to  dollars. 

Ans.  2976  dollars. 


UNITED    STATES    MONEY.  133 

9.  Reduce  249600  cents  to  dollars. 

Ans.  249C  dollars. 
10.  Reduce  22G43  cents  to  dollars. 

Ans.  226.43  dollars. 

The  answer  to  the  last  example  is  read :  Two 
hundred  and  twenty-six  dollars  4  dimes  and  3  cents, 
or  43  cents. 

LESSON    XI. 

SEDUCTION   XI. 

142.  To  reduce  dimes  to  dollars. 

EULE. 
Divide  by  10, 

To  divide  by  10,  cut  off  1  figure  from  the  right. 
10  dimes  make  1  dollar. 

EXAMPLES. 

1.  How  many  dollars  in  200  dimes  ? 

OPERATION. 

200  ~  10  =.20  dollars. 

2.  In  6000  dimes  how  many  dollars  ? 

Ans.  600  dollars. 

3.  In  8000  dimes  how  many  dollars? 

Ans.  800  dollars. 

4.  In  98643  dimes  how  many  dollars  ? 

Ans.  9864.3  dollars. 
12 


134  INTERMEDIATE   ARITHMETIC. 

LESSON  XII. 

ADDITION    OF    UNITED    STATES    MONEY. 

143.  To  add  United  States  money,  proceed  as 
in  simple  addition. 

RULE. 

Write  the  dollars,  cents,  and  mills,  so  that  the 
units  of  the  same  denomination  will  stand  under  each 
other.  Add  as  in  simple  numbers,  and  place  the 
period  under  the  periods  above. 

144.  The  proof  of  addition  of  United  States 
money  is  the  same  as  that  of  simple  addition. 

EXAMPLES. 

1.  Add  $1.253,  $3.28,  $4.962,  $21.38. 

OPERATION. 
1.253 

3.280 

4.962 

21.380 

Ans.      830.875 

145.  NOTE. — In  the  second  and  fourth  lines  a  0  was  written 
to  supply  the  place  of  the  mills.  This  0  is  of  no  value,  and  may 
be  omitted.  It  was  written  to  fill  up  the  space. 

The  above  answer  is  read  :  Thirty  dollars  eighty- 
seven  cents  and  five  mills.  5  mills  is  fV,  or  &,  of  a 
cent. 


UNITED   STATES   MONEY.  135 

2.  Add  $12.3,  $12.34,  $12.345.  Ans.  $36.985. 

3.  Add  $2.345,  $23.45,  $2.345.  Ans.  $28.14. 

4.  Add  $3.456,  $34.56,  $345.6.  Ans.  $383.616. 

5.  Add  $456.7,  $45.67,  $4.567.  Ans.  $506.937. 

6.  Add  $567.8,  $5678,  $5.678.  Ans.  $6251.478. 

7.  Add  $67.89,  $6.789,  $6789.  Ans.  $6863.679. 

8.  Add  $7.891,  $78.91,  $789.1.  Ans.  $875.901. 

9.  Add  $8912,  $7.891,  $891.2.  Ans.  $9811.091. 

10.  Add  $12345.6,  $123.456.    Ans.  $12469.056. 

11.  Add  $23.4567,  $234.567,  $2345.67. 

Ans.  $2603.6937. 

12.  Add  $345.678,  $3.45678,  $345678. 

Ans.  $346027.13478. 

13.  Add  $4.56789,  $4567.89,  $45678.9. 

Ans.  $50251.35789. 

14.  Add  $576.43,  $5764.3,  $5.7543. 

Ans.  $6346.4943. 

15.  Add  $12.3,  $1.23,  $.123,  $123,  $12.3. 

Ans.  $148.953. 

16.  Add  $234,  $23.4,  $2.34,  $.234,  $234. 

Ans.  $493.974. 

17.  Add  $34.5,  $3.45,  $.345,  $345,  $3.45. 

Ans.  $386,745. 

18.  Add  $456,  $4.56,  $456,  $45.6,  $45.6. 

Ans.  $1007.76. 

19.  Add  $56.78,  $56.78,  $56.78,  $56.78,  $56.78. 

Ans.  $283.90. 

20.  Add  $678.9,  $678.9,  $678.9,  $678.9,  $678.9. 

Ans.  $3394.5. 

21.  Add  $78.91,  $78.91,  $78.91,  $78.91,  $78.91. 

Ans.  $394.55. 


136  INTERMEDIATE   ARITHMETIC. 

22.  Add  $8.912,  $8.912,  $8.912,  $8.912,  $8.912. 

Ans.  $44.560. 

23.  Add  $91.23,  $91.23,  $91.23,  $91.23,  $91.23. 

Ans.  $456.15. 

24.  Add  $246.89,  $246.89,  $246.89,  $246.89. 

Ans.  $987.56. 

25.  Add  $46.892,  $46.892,  $46.892,  $46.892. 

Ans.  $187.568. 

26.  Add  $9.6842,  $9.6842,  $9.6842,  $9.6842. 

Ans.  $38.7368. 

27.  Add  $2.4869,  $2.4869,  $2.4869,  $2.4869. 

Ans.  $9.9476. 

28.  Add  $48.692,  $48.692,  $48.692,  $48.692. 

Ans.  $194.768. 

29.  Add  $29.684,  $29.684,  $29.684,  $29.684. 

Ans.  $118.736. 

30.  Add  $123.4567,$123.4567,$123.4567,$123.4567. 

Ans.  $493.8268. 

31.  Add  $21.43567,$21.43567,$21.43567,$21.43567. 

Ans.  $857.4268. 

32.  Add$41.25637,$41.25637,$41.25673,$41.25673. 

Ans.  $165.02620. 

33.  Add  $189,  $981,  $918,  $819,  $198. 

Ans.  $3105. 

34.  Add  $24.68,  $24.68,  $24.68,  $24.68,  $24.68. 

Ans.  $123.40. 

35.  Add  $4.268,  $4.268,  $4.268,  $4.268,  $4.268. 

Ans.  $21.340. 

36.  Add  $862.4,  $862.4,  $862.4,  $862.4.  $362.4. 

Ans.  $4312. 

37.  Add  $.2684,  $.2684,  $.2684,  $.2684,  $.2682. 

Ans.  $1.2418. 


UNITED   STATES   MONEY.  137 

38.  Add  $68.42,  $68.42,  $68.42,  $68.42,  $68.42. 

Ans.  $342.10. 

39.  Add  $28.64,  $28.64,  $28.64,  $28.64,  $28.64. 

Ans.  $143.20. 

40.  Add   two   dollars   eighty-six   cents  and  four 
mills,  twenty-eight  dollars  and  sixty-four  cents,  two 
hundred  and  eighty-six  dollars  and  four  dimes,  two 
thousand  eight  hundred  and  sixty-four  dollars. 

Ans.  $3181.904. 

41.  I  bought  a  slate  for  seventy-five  cents,  one 
grammar  for  one  dollar  and  five  cents,  one  history 
for  two  dollars  and  ten  cents,  one  speller  for  thirty- 
six  cents  and  five  mills.     How  much  did  I  have  to 
pay.  Ans.  $4.265. 

LESSOX  XIII. 

SUBTRACTION    OF   UNITED   STATES   MONEY. 

RULE. 

I4O.  Write  the  numbers  so  that  the  figures  of  the 
same  denomination  will  be  found  under  each  other, 
and  subtract  as  in  simple  numbers. 

]  47.  The  proof  of  subtraction  of  United  States 
money  is  the  same  as  that  of  simple  numbers. 

EXAMPLES. 

1.  From  $29.62  take  $13.025. 

OPERATION. 

29.620 
13.025 


Ans.     $16.595 
12* 


138 


INTERMEDIATE   ARITHMETIC. 


14:8.  It  will  be  seen  from  the  preceding  exam- 
ple that  the  minuend  has  no  mills.  Supply  the 
deficiency  by  annexing  a  0  for  the  mills.  In  prac- 
tice, the  0  is  supposed  to  occupy  the  place.  The 
same  remarks  apply  to  the  subtrahend. 

2.  From  $615.81  take  $615.81.  Am.  0. 

3.  From  $1919.7  take  $191.97.      Ans.  $1727.73. 

4.  From  $718.62  take  $718.62.  Ans.  0. 

5.  From  $1991.7  take  $199.17.      Ans.  $1792.53. 

6.  From  $9109.17  take  $910.917.J.?is.  $8198.253. 

7.  From  $950.99  take  $950.99.  Ans.  0. 

8.  From  $1876.42  take  $1 87.642. Ans.  $1688.778. 

9.  From  $987.342  take  $987.342.  Ans.  0. 

10.  From  $8937.24  take  $8937.24.  Ans.  0. 

11.  From  $9837.24  take  $9837.24.  Ans.  0. 

12.  From  $4983.27  take  $498.327. 

Ans.  $4484.943. 

13.  From  $287.643  take  $287.643.       Ans.  0. 

14.  From  $89726.4  take  $897.264. 

Ans.  $88829.136. 

15.  From  $82764.3  take  $8276.43.^^5.  $74487.87. 

16.  From  $98728.43  take  $9872.843. 

Ans.  $88855.587. 

17.  From  $8764.321  take  $8764.321.  Ans.  0. 

18.  From  $98743.6  take  $9874.36.  Ans. $88869.24. 

19.  From  $89724.3  take  $897.243. 

Ans.  $88627.057. 
W.  From  $87643.28  take  $8764.328. 

Ans.  $78878.952. 


UNITED    STATES   MONEY.  139 

LESSON  XIV. 

21.  I  received  $284.63.     I  owe  I.  Browne  $81.02, 
and  Samuel  Smith  $98.34.     How  much  have  I  left  ? 

Ans.  $105.27. 

22.  The  pay-roll   of   my   laborers    amounts    to 
$867.45.     If  I  draw  one  thousand  dollars  from  bank, 
how  much  will  I  have  remaining  after  paying  the 
roll?  Ans.  $132.55. 

23.  From  200  dollars  I  gave  James  Smith  $25, 
Wm.  Burns,   $28.05,    and    James  Murphy,  $26.45. 
How  much  have  I  left  ?  Ans.  $20.50. 

24.  A  lady  went  to  town  to  make  some  purchases. 
She  bought  one  shawl  for  twenty-five  dollars,  two 
dresses  for  twelve   dollars   each,  twenty  pairs  of 
gloves,  @  two  dollars  each.     Her  father  gave  her 
two    hundred   and    fifty   dollars   for   her   birthday 
present.     How  much  had  she  left  after  buying  the 
above  bill  of  goods?  Ans.  $161. 

25.  A  little  boy  who  was  the  best  scholar  in  his 
class,  was  presented  with  the  following  books  for  his 
devotion  to  study:  One  grammar,  worth  fifty-five 
cents ;  one  history,  worth  one  dollar  and  fifty  cents ; 
one  Scholar's  Companion,  worth  eighty  cents ;  one 
arithmetic,   worth    eighty    cents ;    one    geography, 
worth  one  dollar  and  fifty  cents  ;  one  reader,  worth 
ninety  cents ;  one  speller,  worth  twenty  cents ;  one 
slate,  worth  twenty  cents ;  one  French  book,  worth 
fifty-five  cents.     How  much   did  his  set  of  books 
cost  ?    How  much   change   did  his  father  receive 
from  a  ten-dollar  bill  which  he  gave  in  payment  for 
the  books.  Ans. 


140 


INTERMEDIATE  AEITHMETIC. 


LESSON  XV. 

MULTIPLICATION   OF   UNITED    STATES   MONEY 
KULE. 

149.  Multiply  as  in  simple  numbers.     Cut  of 
as  many  figures  from  the  right  of  the  product  as  there 
are  figures  to  the  right  of  the  decimal  point  in  the 
multiplicand. 

EXAMPLES. 

1.  Multiply  $2.485  by  3. 

150.  1st.  Multiply  2.485  by  3  as  OPERATION. 
in  simple  numbers.     2d.    Count  the 

number  of  figures  in  the  multiplicand 

on  the  right  of  the  decimal  point.   In 

the  above  example  there  are  3  figures.  $7.455 

Cut   off  3   figures   from  the    result, 

counting  from  right  to  left.     The  answer  is  7  dollars 

45  cents  and  5  mills. 


2.  Multiply  $1234.5  by  12. 
3.  Multiply  8234.56  by  21. 
4.  Multiply  £67.891  by  13. 
5.  Multiply  $7.8912  by  32. 
6.  Multiply  $8912.3  by  14. 
7.  Multiply  $912.34  by  46. 
8.  Multiply  $43.219  by  21. 
9.  Multiply  $3.4219  by  18. 
10.  Multiply  $4319.2  by  97. 
11.  Multiply  $291.34  by  10. 
12.  Multiply  $67.289  by  28. 

Ans.  $14814. 
Ans.  $4925.76. 
Ans.  $882.583. 
Ans.  $252.5184. 
Ans.  $124772.2. 
Ans.  $41967.64. 
Ans.  $907.599. 
Ans.  $61.5942. 
Ans.  $418962.4. 
Ans.  $2913.40. 
Ans.  $1884.092. 

UNITED   STATES   MONEY.  141 

13.  Multiply  $7.6829  by  19.  Am.  $145.9751. 

14.  Multiply  $9762.8  by  98.  Am.  $956754.4. 

15.  Multiply  $079.28  by  18.  Ans.  $12227.04. 

16.  Multiply  $76.928  by  27.  Am.  $2077.056. 

17.  Multiply  $8.9267  by  86.  Ans.  $767.6962. 

18.  Multiply  $2438.7  by  14.  Ans.  $34141.8. 

19.  Multiply  $23.486  by  36.  Am.  $845.496. 


LESSON  XVI. 

20.  I   sold   26   bushels   of  potatoes,   at   $2.45   a 
bushel.     How  much  was  the  bill  ?  Am.  $63.70. 

21.  I  bought  from  Jas.  Brown  12  sacks  of  bran, 
at   $3.40  per  sack.     What  was  the  amount  of  the 
bill?  Am.  $40.80. 

22.  James  Smith  bought  2  hogs,  each  weighing 
96  pounds,  at  twelve  cents  a  pound.     How  much 
were  they  worth?  Ans.  $23.04. 

23.  Jos.  Brown  sold  20  grammars,  worth  $1.50 
each,  and  30  spellers,  worth  25  cents  each.     What 
was  the  amount  of  the  sale?  Ans.  $37.50. 

24.  What  is  the  value  of  2  hogsheads  of  sugar, 
each  weighing  900  pounds,  at  14  cents  per  pound? 

Ans.  $252. 

25.  What  will  45  acres  of  land  cost,  at  $3.375  per 
acre?  Ans.  $151.875. 

26.  What  will  240  yards  of  cloth  cost,  at  two 
dollars  forty-five  cents  and  five  mills  per  yard  ? 

Ans.  $589.20. 


142  INTERMEDIATE   ARITHMETIC. 

27.  I  employ  40  men  for  20  days,  at  $2.25  per 
day.     How  much  must  I  pay  ?  Ans.  $2000. 

28.  I  received  2  loads  of  lumber,  each  load  2500 
feet,  at  2  cents  and  5  mills  per  foot.     How  much 
was  it  worth?  Ans.  $125. 

29.  A  merchant  sold  24  barrels  of  beef,  each  bar- 
rel weighing  196  pounds,  at  11  cents  and  5  mills  per 
pound.     What  did  he  receive  ?  Ans.  $540.96. 


LESSOR  XVII. 

DIVISION    OF    UNITED    STATES   MONEY. 
RULE. 

151.  Divide  as  in  simple  numbers.     From  t) 
right  of  the  quotient  cut  off  (counting  from  right  to 
left],  as  many  figures  as  there  are  figures  in  the  divi- 
dend to  the  right  of  the  decimal  point.     The  figures  in 
the  quotient  to  the  left  of  the  decimal  point  represent 
dollars  /  those  on  the  right  of  the  decimal  point  rep- 
resent dimes,  cents,  mills,  etc.,  according  to  the  posi- 
tion they  occupy. 

152.  The  proof  of  division  is  multiplication. 

153.  NOTE. — If  the  dividend  is  dollars,  and  is  smaller  than 
the  divisor,  reduce  it  to  cents,  and  divide.  In  this  case,  the  answer 
would  be  cents.  If  not  large  enough  when  reduced  to  cents,  reduce 
the  cents  to  mills  and  divide.  The  answer  in  this  case  would  be 
mills. 


UNITED  STATES  MONEY. 


143 


EXAMPLES. 


1.  Divide  $25.50  among  5  men.     How  much  will 
each  man  receive  ? 


OPERATION. 

5  )  25.50 
5.10 


154.  Divide  as  in  division  of 
simple  numbers.  Count  the  number 
of  figures  to  the  right  of  the  decimal 
point  in  the  dividend,  and  cut  off  as 
many  in  the  quotient,  counting  from,  right  to  left. 
Tho  quotient  in  the  above  is  510.  The  number  of 
figures  in  the  dividend  to  the  right  of  the  decimal 
point  is  2.  By  cutting  off  2  figures  from  the  quo- 
tient, the  answer  will  be,  5  dollars  and  10  cents. 


PROOF. 
Quotient.          Divisor.          Dividend. 

5.10      X      5      =      25.50 


2.  Divide  $1.50  between  5  boys. 

$1.50  is  equal  to  one  hundred 
and  fifty  (150)  cents.  The  ques- 
tion is  now,  Divide  150  cents  be- 
tween 5  boys.  5  into  150, 30  times, 
or  30  cents  each  boy. 


OPERATION. 


5)1.50 


30  cents. 


155.  When  the  highest  denomination  of  the 
dividend  is  not  divisible  by  the  divisor,  consider  the 
dividend  reduced  to  the  lowest  denomination,  and 
divide. 


144  INTERMEDIATE   ARITHMETIC. 

3.  Divide  $123.45  by  5.  Ans.  $24.69. 

4.  Divide  $234.56  by  2.  Ans.  $117.28. 

5.  Divide  $345.670  by  10.  Ans.  $34.567. 

6.  Divide  $4.5678  by  2.  Ans.  $2.2879. 

7.  Divide  $5678.90  by  5.  Ans.  $1135.78. 

8.  Divide  $6789.10  by  10.  Ans.  $678.91. 

9.  Divide  $78.912  by  2.  Ans.  $39.456. 

10.  Divide  $891.230  by  5.  Ans.  $178.246. 

11.  Divide  $9123.4  by  2.  Ans.  $4561.7. 

12.  Divide  $123.456  by  2.  Ans.  $61.728. 

13.  Divide  $5643.210  by  5.  Ans.  $1128.642. 

14.  Divide  $678.90  by  5.  Ans.  $135.78. 

15.  Divide  $5432.80  by  5.  Ans.  $1086.56. 

16.  Divide  $7643.200  by  100.  Ans.  $76.432. 

17.  Divide  $97654.30  by  10.  Ans.  $9765.43. 

18.  Divide  $46.800  by  100.  Ans.  $.468. 

19.  Divide  $96.8700  by  100.  Ans.  $.9687. 

20.  Divide  $247.250  by  10.  Ans.  $24.725. 

21.  Divide  $3376.420  by  10.  Ans.  $337.642. 

22.  Divide  $286.7310  by  10.  Ans.  $28.6731. 

23.  Divide  $98764.20  by  5.  Ans.  $19752.84, 

24.  Divide  $1876.5400  by  100.  Ans.  $18.7654. 


UNITED   STATES   MONEY.  145 


LESSON  XVIII. 

25.  If  25  acres  of  land  cost  $125.25,  what  is  the 
cost  of  1  acre  ?  Ans. 

20.  If  18  barrels  of  apples  cost  99  dollars,  what  is 
1  barrel  worth  ?  Ans. 

27.  If  6  cords  of  wood  cost  $28.50,  what  is  1  cord 
worth?  Ans.  $0.75. 

28.  If  20  barrels  of  beef  cost  $375,  what  is  1  bar- 
rel worth?  Ans.  $18.75. 

29.  A  merchant  paid  $5000.25  for  25  bales  of  cot- 
ton.    What  was  the  price  of  1  bale  ?    Ans.  $200.25. 

30.  A   grocer  pays   $31.25   for  250   pounds    of 
cheese.     What  is  1  pound  worth  ?      Ans.  125  mills. 

31.  I  gave  250  dollars  for  25  gallons  of  whiskey. 
How  much  is  the  whiskey  per  gallon  ? 

Ans.  10  dollars. 

32.  I  paid  $110.25  for  03  pounds  of  tea.     How 
much  is  1  pound  worth?  Ans.  $1.75. 

33.  Sam.  Browne  bought  240  gallons  of  wine,  for 
which  he  paid  $444.     How  much  was  the  wine  worth 
per  gallon?  Ans.  $1.85. 

34.  If  33  sheep  cost  $92.50,  what  is  the  price  of  1 
sheep  ?  Ans.  $2.50. 

35.  If  GO  barrels  of  cider  cost  $144,  how  much  is 
1  barrel  worth  ?  Ans.  $2.40. 

13 


146 


INTERMEDIATE  ARITHMETIC. 


CHAPTER  VIII. 

QUESTIONS  BY  ANALYSIS. 
LESSOR  I. 

156.  ANALYSIS  is  the  solution  of  a  problem, 
step  by  step,  resolving  it  into  parts,  thereby  render- 
ing it  more  plain  and  intelligible. 

157.  The  price  of  one  article  being  given,  to 
find  the  value  of  several  articles. 

RULE. 

Multiply  the  number  of  articles  or  things  by  the 
price  of  one  of  those  articles  or  things. 

EXAMPLES. 

1.  If  1  pound  of  beef  costs  4  cents,  how  much 
•will  20  pounds  cost  ? 

ANALYSIS. 

158.  Since  1  pound  is  worth  4  cents,  20  pounds 
are  worth  20  times  4  cents,  or  80  cents. 

2.  If  1  apple  costs  2  cents,  what  will  20  apples 
cost  ?  Ans.  40  cents. 

3.  If  1   pound  of  meat  costs  12  cents,  what  will 
12  pounds  cost  ?  Ana.  144  cents. 

4.  If  1  ton  of  hay  costs  22  dollars,  what  will  8 
tons  cost?  Ans.  $176. 


ANALYSIS.  147 

5.  If  1  hat  costs  $8.75,  how  much  will  20  hats 
cost  ?  25  hats  ?  4  hats  ?  40  hats  ?  3  hats  ? 

Last  Ans.  $26.25. 

G.  If  1  barrel  of  flour  is  worth  $6.50,  how  much  are 
2  barrels  worth?  3  barrels?  4  barrels  ?  5  barrels? 
6  barrels  ?  7  barrels  ?  8  barrels  ?  Last  Ans.  $52.00. 


LESSON  II. 

1«>9.  The  number  of  articles  and  the  price  of  all 
of  them  being  given,  to  find  the  price  of  one  article. 

RULE. 
Divide  the  cost  of  all  ~by  the  number  of  things. 

EXAMPLES. 

1.  If  150  bushels  of  corn  cost  $103.50,  what  is  1 
bushel  worth  ? 


Since  150  bushels  are 
worth  $10.3.50,  1  bushel 
is  worth  1  one  hundred 
and  fiftieth  part 
$103.50,  which  is 
cents. 


OPERATION. 


150  )  103.50  (  69  cents. 


900 


1350 
1350 


2.  Bought  650  barrels  of  flour  for  4225  dollars. 
What  is  the  price  of  1  barrel?  of  150  barrels? 

Last  Ans.  975  dollars. 

3.  If  I  paid  $20250  for  450  acres  of  land,  what  is 
1  acre  worth  ?  Ans.  $4.50. 


148 


INTERMEDIATE   ARITHMETIC. 


4.  Mrs.  Brown  paid   125   dollars  for  25  dresses. 
What  is  cost  of  1  dress  ?  Am.  5  dollars. 

5.  If  25   bushels  of  potatoes  can  be  bought  for 
$12.50,  how  much  is  1  bushel  worth?  Am.  50  cents. 

6.  If  2  dozen  of  hats  cost  $4.80,  how  much  is  1 
hat  worth  ?  Am.  20  cents. 

7.  Samuel  Jones  bought  "10  spellers  for  $2.50. 
What  was  the  price  of  1  speller  ?  Ans.  25  cents. 

8.  Seventeen   boys   contribute    $85.85    towards 
defraying  the  expenses  of  a  picnic.     How  much  did 
each  boy  contribute?  Ans.  $5.05. 

9.  Robert  bought  250  readero  and  paid  $62,50  for 
them.     How  much  did  each  reader  cost  ? 

Ans.  25  cents. 

10.  A  school  requires  100  grammars,  the  cost  of 
which  is  $55.     What  is  1  grammar  worth  ? 

Ans.  55  cents. 

11.  The  second  class  of  the  same  school  require 
50  grammars,  the  cost  of  which  is  $7.50.     What  is 
the  cost  of  1  grammar?  Ans.  15  cents. 

12.  If  10  histories  cost  $15,  what  is  the  price  of  1 
history?  Ans.  $1.50. 

13.  If  20  arithmetics  cost  $14,  what  is  the  value 
of  1  arithmetic  ?  Ans.  70  cents. 

14.  If  250  large  geographies  cost  $375,  how  much 
is  1  geography  worth  ?  Ans.  $1.50. 

15.  A  grocer  sold  20  pounds  of  peas  for  $3.00. 
What  is  1  pound  worth?  Ans.  15  cents. 

16.  I  received  for  rent  of  my  house,  for  1  year, 
$330.     How  much  per  month  did  I  rent  my  house 
for  ?     How  much  per  day  ?  First  Ans.  $27.50. 


ANALYSIS.  149 

17.  If  a  man  spends  $1500  a  year  for  the  support 
of  his  house,  how  much  does  he  spend  per  month? 

Am.  $125. 
How  much  per  day  ?  Ans. 

18.  A  gentleman  receives  $3000  a  year  for  his 
salary,  J  of  which  he  puts  aside  and  saves  ;  the  bal- 
ance he  spends.     How  much  does  he  spend  every 
month  ?  Ans. 


LESSON  III. 

I6O.  The  price  of  several  articles  and  the  price 
of  one  being  given,  to  find  the  number  of  articles. 

KULE. 

Divide  the  price  of  all  of  the  articles  by  the  price 
of  one  article.  The  quotient  will  be  the  answer. 

EXAMPLES. 

1.  How  much  sugar  can  I  buy  for  $125,  if  sugar 
is  worth  125  mills  a  pound? 

ANALYSIS. 

First  reduce  the  dollars  to  mills.  1000  mills 
make  1  dollar.  In  125  dollars  there  will  be  125 
times  1000.  To  multiply  any  number  by  1000, 
annex  three  O's.  The  sum  can  now  be  read  :  How 
many  pounds  of  sugar  can  I  buy  for  125000  mills,  if 
sugar  is  worth  125  mills  per  pound.  Divide  125000 
by  125,  the  answer  will  be  pounds. 

125000  -f-  125  =  1000  pounds. 
13* 


150  INTERMEDIATE  ARITHMETIC. 

PROOF. 

1000  Ibs.  X  .125  —  $125.000 

Take  off  3  figures  from  the  right  for  the  mills,  or 
divide  the  product  by  1000,  which  is  the  same  as 
taking  off  3  figures  from  the  right. 

2.  If  coal  was  worth  10  dollars  a  load,  how  many 
loads  could  I  buy  for  100  dollars  ?     Why  ? 

Ans.  10  loads. 

3.  If  I  pay  $103.50  for  corn,  at  96  cents  a  bushel, 
how  many  bushels  do  I  buy?  Ans.  150  bushels. 

4.  If  flour  is  worth  $6.50  a  barrel,  how  many  bar- 
rels can  I  buy  for  4225  dollars  ?        Ans.  650  barrels. 

5.  If  land  is  worth  $4.50  an  acre,  how  many  acres 
can  I  buy  for  20250  dollars  ?  Ans.  450  acres. 

6.  If  1  dress  costs  5  dollars,  how  many  dresses 
can  I  buy  for  125  dollars  ?  Ans.  25  dresses. 

7.  If  1  hat  costs  20  cents,  how  many  hats  can  I 
buy  for  $4.80  ?  Ans.  2  doz.  or  24  hats. 

8.  If  1  speller  is   worth    25   cents,   how  many 
spellers  can  I  buy  for  $2.50  ?  Ans.  10  spellers. 

9.  A  number  of  boys  contribute  $85.85  towards 
defraying  the  expenses  of  a  picnic.     Each  boy  con- 
tributes $5.05.     How  many  boys  contributed  ? 

Ans.  17  boys. 

10.  If  1  reader  costs  25  cents,  how  many  readers 
can  be  bought  for  $62.50  ?  Ans.  250. 

11.  If  1  grammar  is  worth   55  cents,  how  many 
grammars  can  be  purchased  for  55  dollars  ?  Ans.  100. 

12.  If  a  person  spends  125  dollars  a  month,  how 
many  months  will  1500  dollars  last?  Ans.  12  months. 


PROPERTIES   OF   NUMBERS.  151 


CHAPTER  IX. 

PROPERTIES  OF  NUMBERS. 

LESSON  I. 

161.  AN  INTEGRAL  NUMBER  is  a  whole  number. 
1 63.  An  even  number  is  one  which  is  exactly 
divisible  by  2  ;  as,  4,  6,  8,  10,  etc. 

163.  An  odd  number  is  one  which  is  not  exactly 
divisible  by  2;  as,  3,  5,  9,  15,  etc. 

164.  A  prime  number  is  one  which  is  divisible 
by  no  number  but  itself  and  1 ;  as,  5,  7,  11,  13. 

163.  No  even  number  can  be  a  prime  number, 
except  the  figure  2. 

166.  The  prime  numbers  between  1  and  100 
are:  1,  2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  37,  41,  43, 
47,  53,  59,  61,  67,  71,  73,  79,  83,  89,  97. 

167.  A  composite  number  is  the  product  of  two 
or  more  prime  numbers;    thus,  12  is  a  composite 
number,  because  it  is  the  product  of  2X2X3,  which 
are  prime  numbers. 

168.  Any  composite  number  is  exactly  divisible 
by  any  of  its  prime  numbers,  or  by  the  product  of 
several  of  its  prime  numbers;  thus,  12  is  divisible  by 
3,  one  of  its  prime  factors,  or  by  3X2=6,  the  pro- 
duct of  two  of  its  prime  factors. 

169.  When  two  numbers   are  so  constituted 


152  INTERMEDIATE   AKITHMET1G'. 

that  they  cannot  be  exactly  divided  by  the  same 
number.,  the  two  numbers  are  said  to  be  prime  to 
each  other;  thus,  36  and  17  are  prime  to  each  other, 
because  no  number  will  exactly  divide  both  of  them 
without  a  remainder. 

1  70.  Every  number  is  equal  to  the  product  of 
its  prime  numbers.  For  example,  32,  the  prime 
numbers  of  which  are  2X2X2X2X2. 

171.  If  we  multiply  the  prime  numbers  togeth- 
er, their  product  must  be  equal  to  the  composite 
number. 

17S.  A  common  divisor  between  iwo  or  more 
numbers  is  a  number  which  will  exactly  divide 
them;  thus,  3  is  a  common  divisor  of  6,  12,  18,  24. 

173.  Any  set  of  numbers  which  are  prime  to 
each  other,  have  no  common  divisor  except  unity. 

1 74.  The  greatest  common  divisor  is  the  greatest 
number  which  will  divide  a  series  of  numbers  with- 
out a  remainder;  thus,   6  is  the  greatest  common 
divisor  for  6,  12,  18,  24,  36,  etc.,  because  it  is  the 
largest  number  which   will  divide  them  without  a 
remainder. 

1 75.  A  multiple  is  the  product  of  two  or  more 
prime  numbers ;  thus,  6  is  a  multiple,  whose  prime 
factors  are  2  and  3. 

176.  A  common  multiple  is  a  number  which  will 
exactly  contain  a  series  of  numbers  without  a  remain- 
der; thus,   24  is  a  common  multiple  for  6  and  12, 
because   it    will   contain  both  of   them   without   a 
remainder. 

177.  The  least  common  multiple  for  two   or 


DIVISIBILITY   OF   NUMBERS.  153 

more  numbers  is  the  smallest  number  which  will 
contain  them  without  a  remainder ;  thus,  24  is  the 
least  common  multiple  for  6,  12,  and  24,  because  no 
smaller  number  will  contain  every  one  of  them  with- 
out a  remainder. 

IT 8.  Every  number  must  be  the  least  common 
multiple  of  each  and  every  one  of  its  prime  numbers. 


DIVISIBILITY  OF  NUMBERS. 

179.  Any  number  ending  in  0,  2,  4,  6,  or  8,  is 
divisible  by  2. 

1 8O.  Any  number  the  sum  of  whose  figures  is 
divisible'by  3,  can  be  divided  by  3. 

181.  Four  will  divide  any  number  whose  two 
last  figures  can  be  divided  by  4. 

182.  Four  will  divide  any  number  ending  in 
two  or  more  O's. 

183.  Any  number  ending  in  0  or  5  is  divisible 
by  5. 

184.  Five  will  divide  any  number  ending  in  one 
or  more  O's. 

1 85.  Ten  divides  any  number  ending  in  0. 

186.  Any  number,  the  sum  of  whose  figures  is 
divisible  by  3  or  9,  can  be  divided  by  3  or  9. 


154 


INTERMEDIATE   ARITHMETIC. 


FACTORING. 

LESSON    II. 

1 87.  To  find  the  prime  factors  of  any  number. 

RULE. 

Divide  the  number  by  the  smallest  prime  number 
which  will  exactly  divide  it.  Divide  the  quotient  by 
the  smallest  prime  number  which  will  exactly  divide  it 
without  a  remainder.  Divide  as  before  until  the  quo- 
tient becomes  1.  The  divisors  are  the  prime  numbers. 

EXAMPLES. 

1.  Find  the  prime  factors  of  1800. 

OPEEATION. 

2)  1800 

2  )  900 

2  )  450 

5  )  225 

5  )  45 

3~yir 

3  )  3 
1 

The  prime  factors  are  2,  2,  2,  5,  5,  3,  3,  1. 

2.  Find  the  prime  factors  of  200. 

Ans.  2,  2,  2,  5,  5. 

3.  Find  the  prime  factors  of  250. 

Ans.  2,  2,  2,  2,  3,  5. 


FACTORING.  155 

4.  Find  the  prime  factors  of  300. 

Ans.  2,  2,  3,  5,  5. 

5.  Find  the  prime  factors  of  350.  Ans.  2,  5,  5,  7. 
C.  Find  the  prime  factors  of  400. 

Ans.  2,  2,  2,  2,  5,  5. 

7.  Find  the  prime  factors  of  450. 

Ans.  2,  5,  5    3    3. 

8.  Find  the  prime  factors  of  500. 

Ans.  2,  2,  5,  5,  5. 

9.  Find  the  prime  factors  of  550. 

Ans.  2,  5,  5,  11. 

10.  Find  the  prime  factors  of  GOO. 

Ans.  2,  2,  2,  3,  5,  5. 

11.  Find  the  prime  factors  of  G50. 

Ans.  2,  5,  5,  13. 

12.  Find  the  prime  factors  of  700. 

Ans.  2,  2,  5,  5,  7. 

13.  Find  the  prime  factors  of  750. 

Ans.  2,  3,  5,  5,  5. 

14.  Find  the  prime  factors  of  800. 

Ans.  2,  2,  2,  2,  2,  5,  5. 

15.  Find  the  prime  factors  of  850. 

Ans.  2,  5,  5,  17. 

16.  Find  the  prime  factors  of  900. 

Ans.  2,  2,  3,  3,  5,  5. 

17.  Find  the  prime  factors  of  950. 

Ans.  2,  5,  5,  19. 

18.  Find  the  prime  factors  of  1000. 

Ans.  2,  2,  2,  5,  5,  5. 

19.  Find  the  prime  factors  of  1100. 

Ans.  2,  2,  5,  5,  11. 


156  INTERMEDIATE   ARITHMETIC. 

20.  Find  the  prime  factors  of  1200. 

Ans.  2,  2,  2,  2,  3,  5,  5. 

21.  Find  the  prime  factors  of  1250. 

Ans.  2,  5,  5,  5,  5. 

22.  Find  the  prime  factors  of  1300. 

Ans.  2,  2,  5,  5,  13. 

23.  Find  the  prime  factors  of  1350. 

Ans.  2,  3,  3,  3,  5,  5. 

24.  Find  the  prime  factors  of  1400. 

Ans.  2,  2,  2,  5,  5,  7. 

25.  Find  the  prime  factors  of  1450.  Ans.  2,  5,  5,  29. 

26.  Find  the  prime  factors  of  1500. 

Ans.  2,  2,  3,  5,  5,  5. 

27.  Find  the  prime  factors  of  1550. 

Ans.  2,  5,  5,  31. 

28.  Find  the  prime  factors  of  1600. 

Ans.  2,  2,  2,  2,  2,  2,  5,  5. 

29.  Find  the  prime  factors  of  1650. 

Ans.  2,  3,  5,5,  11. 

30.  Find  the  prime  factors  of  1700. 

Ans.  2,  2,  5,  5,  17. 

31.  Find  the  prime  factors  of  1750. 

Ans.  2,  5,  5,  5,  7. 

32.  Find  the  prime  factors  of  1800. 

Ans.  2,  2,  2,  3,  3,  5,  5. 

33.  Find  the  prime  factors  of  1850. 

Ans.  2,  5,  5,  37. 

34.  Find  the  prime  factors  of  1900. 

Ans.  2,  2,  5,  5,  19. 

35.  Find  the  prime  factors  of  1950. 

Ans.  2,  3,  5,  5,  13. 


FACTOEING.  157 

36.  Find  the  prime  factors  of  2000. 

Ans.  2,  2,  2,  2,  5,  5,  5. 

37.  Find  the  prime  factors  of  2100. 

Ans.  2,  2,  3,  6,  5,  7. 

38.  Find  the  prime  factors  of  2150. 

Ans.  2,  5,  5,  43. 

39.  Find  the  prime  factors  of  2200. 

Ans.  2,  2,  2,  5,  5,  11. 

40.  Find  the  prime  factors  of  2250. 

Ans.  2,  3,  3,  5,  5,  5. 

41.  Find  the  prime  factors  of  3000. 

Ans.  2,  2,  2,  3,  5,  5,  5. 

42.  Find  the  prime  factors  of  3150  . 

Ans.  2,  3,  3,  5,  5,  7. 

43.  Find  the  prime  factors  of  3250. 

Ans.  2,  2,  5,  167. 

44.  Find  the  prime  factors  of  3350. 

Ans.  2,  5,  5,  67. 

45.  Find  the  prime  factors  of  4000. 

Ans.  2,  2,  2,  2,  3,  5,  17. 

46.  Find  the  prime  factors  of  4500. 

Ans.  2,  5,  5,  91. 

47.  Find  the  prime  factors  of  8050. 

Ans.  2,  5,  5,  161. 

48.  Find  the  prime  factors  of  9050. 

Ans.  2,  5,  5,  181. 


14 


158  INTERMEDIATE  ARITHMETIC. 

GREATEST  COMMON  DIYISOR. 
LESSON  IIL 

188.  To  find  the  greatest  common  divisor  be- 
tween two  numbers : 

ETJLE. 

Divide  the  greater  number  ~by  the  less.  If  there  is 
a  remainder,  make  the  remainder  the  divisor,  and  the 
divisor  the  dividend.  If  there  is  a  remainder  again, 
make  the  last  remainder  the  divisor  and  the  first  re- 
mainder the  dividend.  Proceed  in  this  way  until 
there  is  no  remainder.  The  last  divisor  is  the  great- 
est common  divisor. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  24 
and  36  ? 

OPEEATION". 

24  )  36  (  1 
24 

12  )  24  (  2 
24 

12  is  the  greatest  common  divisor. 

2.  What  is  the  greatest  common  divisor  of  6  and 
12?  A?is.Q. 

3.  What  is  the  greatest  common  divisor  of  12 
and  24?  Ans.  12. 


GREATEST   COMMON   DIVISOR.  159 

4.  What  is  the  greatest  common  divisor  of  24 
and  36  ?  Ans.  12. 

5.  What  is  the  greatest  common  divisor  of  18 
and  40  ?  Ans.  2. 

6.  What  is  the  greatest  common  divisor  of  21 
and  30  ?  Ans.  3. 

7.  What  is  the  greatest  common  divisor  of  30 
and  45  ?  Ans.  15. 

8.  What  is  the  greatest  common  divisor  of  15 
and  40  ?  Ans.  5. 

9.  What  is  the  greatest  common  divisor  of  16 
and  32.  Ans.  16. 

10.  What  is  the  greatest  common  divisor  of  19 
and  38?  Ans.  19. 

11.  What  is  the  greatest  common  divisor  of  25 
and  40  ?  Ans.  5. 

12.  What  is  the  greatest  common  divisor  of  30 
and  21  ?  Ans.  3. 

13.  What  is  the  greatest  common  divisor  of  42 
and  20?  Ans.  2. 

14.  What  is  the  greatest  common  divisor  of  16 
and  30  ?  Ans.  2. 

15.  What  is  the  greatest  common  divisor  of  120 
and  200  ?  Ans.  40. 

16.  What  is  the  greatest  common  divisor  of  200 
and  450  ?  Ans.  50. 

17.  What  is  the  greatest  common  divisor  of  650 
and  700.  Ans.  50. 

18.  What  is  the  greatest  common  divisor  of  780 
and  2900.  Ans.  60. 


160  INTERMEDIATE   ARITHMETIC. 

LESSON  IY. 

189.  When  there  are  more  than  two  numbers. 
EULE. 

I.  Find  the  greatest  common  divisor  of  any  two 
of  them. 

II.  Find  the  greatest  common  divisor  between  the 
divisor  found  and  one  of  the  other  numbers. 

III.  Find  the  greatest  common  divisor  between  the 
last  divisor  and  the  next  number. 

IV.  Proceed  in  this  way  until  all  of  the  numbers 
are  divided. 

V.  The  last  divisor  is  the  greatest  common  divisor 
of  all  the  numbers. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  between 
12,  24,  and  38. 

OPERATION". 

12)24(2  12)38(3 

24  36 

2)  12  (6 

12 

First  find  the  divisor  between  12  and  24,  which  is 
12.  Find  the  divisor  between  12  and  38.  The  last 
divisor,  2,  is  the  greatest  common  divisor. 


GREATEST   COMMON   DIVISOK.  161 

2.  What  is  the  greatest  common  divisor  of  12, 
36,  and  48?  Am. 

3.  What  is  the  greatest  common  divisor  of  18, 
24,  and  30  ?  Am. 

4.  What  is  the  greatest  common  divisor  of  22, 
30,  and  40  ?  Am. 

5.  What  is  the  greatest  common  divisor  of  23, 
46,  and  50  ?  Am.  1. 

6.  What  is  the  greatest  common  divisor  of  2,  5, 
and  7?  Am.  1. 

7.  What  is  the  greatest  common  divisor  of  16, 

18,  and  20  ?  Am.  2. 

8.  What  is  the  greatest  common  divisor  of  20, 
22,  and  24  ?  Am.  2. 

9.  What  is  the  greatest  common  divisor  of  22,  24, 
and  26  ?  Am.  2. 

10.  What  is  the  greatest  common  divisor  of  17, 

19,  and  21  ?  Am.  1. 

11.  What  is  the  greatest  common  divisor  of  13, 
18,  22,  and  26?  Am.  1. 

12.  What  is  the  greatest  common  divisor  of  5,  7, 
9,  and  11?  Am.  1. 

13.  What  is  the  greatest  common  divisor  of  10, 
14,  18,  and  22?  Ans.  2. 

14.  What  is  the  greatest  common  divisor  of  20, 
28,  36,  and  44  ?  Ans.  4. 

15.  What  is  the  greatest  common  divisor  of  40, 
56,  72,  and  88  ?  Ans.  8. 

16.  What  is  the  greatest  common  divisor  of  80, 
112,  144,  and  176  9  Ans.  16. 

14* 


162  INTERMEDIATE   ARITHMETIC. 

17.  What  is  the  greatest  common  divisor  of  12, 
24,  and  36  ?  Ans.  12. 

18.  What  is  the  greatest  common  divisor  of  24, 
48,  and  72  ?  Ans.  24. 

19.  What  is  the  greatest  common  divisor  of  24, 
96,  and  144?  Ans.  24. 

20.  What  is  the  greatest  common  divisor  of  3, 12, 
15,  and  18?  Ans.  3. 

21.  What  is  the  greatest  common  divisor  of  27, 
108,  135,  and  162?  Ans.  27. 


LESSON  Y. 

22.  What  is  the  greatest  common  divisor  of  2,  4, 
and  6  ?  Ans.  2. 

23.  What  is  the  greatest  common  divisor  of  4,  6, 
and  8  ?  Ans.  2. 

24.  What  is  the  greatest  common  divisor  of  8, 10, 
and  12?  Ans.  2. 

25.  What  is  the  greatest  common  divisor  of  12, 
14,  and  16  ?  Ans.  2. 

26.  What  is  the  greatest  common  divisor  of  14, 
16,  and  18?  Ans.  2. 

27.  What  is  the  greatest  common  divisor  of  142, 
421,  and  28?  Ans.  1. 

28.  What  is  the  greatest  common  divisor  of  364, 
472,  and  983?  Ans.  1. 

29.  What  is  the  greatest  common  divisor  of  867, 
1734,  and  3468?  Ans.  867. 

30.  What  is  the  greatest  common  divisor  of  24, 
240,  and  2160  ?  Ans.  24. 


LEAST   COMMON   MULTIPLE.  103 

31.  What  is  the  greatest  common  divisor  of  81, 
405,  and  2430?  Am.  81. 

32.  What  is  the  greatest  common  divisor  of  72, 
648,  and  2592  ?  Ans.  72. 

33.  What  is  the  greatest  common  divisor  of  1800, 
2000,  and  2500?  Ans.  100. 

34.  What  is  the  greatest  common  divisor  of  2100, 
2600,  and  2800  ?  Ans.  100. 

35.  What  is  the  greatest  common  divisor  of  2400, 
1728,  and  1200?  Ans.  48. 

36.  What  is  the  greatest  common  divisor  of  195, 
200,  and  205  ?  Ans.  5. 

37.  What  is  the  greatest  common  divisor  of  218, 
318,  418,  518,  and  286  ?  Ans.  2. 

38.  What  is  the  greatest  common  divisor  of  126, 
248,  428,  648,  and  288  ?  Ans.  2. 


LEAST  COMMON  MULTIPLE. 
LESSON  VI. 

19O.  To  find  the  least  common  multiple  of  two 
or  more  numbers. 

KULE. 

I.  "Write  the  numbers  in  a  row. 

II.  Divide  the  numbers  in  the  row  by  any  prime 
number  which  will  exactly  divide  two  or  more  of  them 
without  a  remainder. 


164  INTERMEDIATE  ARITHMETIC. 

III.  The  numbers  in  the  row  not  divisible  by  the 
factor  used  must  be  written  below,  among  the  quo- 
tients. 

IV.  Divide  the  quotients  and  the  numbers  taken 
down  by  any  prime  number  which  will  exactly  divide 
two  or  more  of  them  without  a  remainder. 

V.  Write  the  quotients  and  the  numbers  not  divis- 
ible by  the  factor  used  below  and  divide  as  before. 

VI.  Continue  in  this  way  until  there  are  no  two 
numbers  exactly  divisible  by  any  prime  factor. 

VII.  Multip7y  all  the  divisors  and  the  numbers 
remaining  undivided,  if  any,  together,  and  the  result 
will  be  the  least  common  multiple. 

NOTE. — The  least   common  multiple  and  the  least  common 
denominator  are  the  same. 

EXAMPLES. 

1.  Find  the  least  common  multiple  of  2,  4,  6,  8, 
and  10. 

OPERATION. 

2  )  2  —  4  —  6  —  8  —  10 

2  )  1  —  2  —  3  —  4 —    5 

1  —  1  —  3  —  2  —    5 

2x2x3x2x5  =  1 20,  the  least  common  multiple. 

2.  Find  the  least  common  multiple  of  3,  6,  and  9. 

Am.  18. 

3.  Find   the   least   common  multiple  of  12,  18, 
and  24.  Am.  72. 

4.  Find  the  least  common1  multiple  of  12,  24, 
and  32.  -Am.  96. 


LEAST   COMMON   MULTIPLE.  165 

5.  Find   the   least   common    multiple   of  8,    16, 
and  48.  Ans.  384. 

6.  Find  the  least  common  multiple  of  9,  18,  and 
36.  Am.  36. 

7.  Find  the  least  common  multiple  of  2, 4,  and  12. 

Ans.  12. 

8.  Find  the  least  common  multiple  of  0,  8,  and  7. 

Am.  504. 

9.  Find  the  least  common  multiple  of  8,  7,  and  14. 

Ans.  56. 

10.  Find  the  least   common   multiple  of   3,    7, 
and  21.  Ans.  21. 

11.  Find   the   least   common   multiple   of    4,    9, 
and  22.  Ans.  396. 

12.  Find   the   least   common    multiple  of  5,   10, 
and  31.  Ans.  310. 

13.  Find  the   least   common   multiple   of    8,   7, 
and  24.  Ans.  168. 

14.  Find  the   least   common   multiple  of    2,    4, 
and  6.  Ans.  12. 

15.  Find  the   least   common   multiple   of  4,  6, 
and  8.  Ans.  24. 

16.  Find  the   least   common   multiple   of  6,    8, 
and  10.  Ana.  120. 

17.  Find  the   least   common   multiple  of  8,  10, 
and  12.  Ans.  120. 

18.  Find  the  least  common  multiple  of  10,  12, 
and  14.  Ans.  420. 


166  INTERMEDIATE   ARITHMETIC. 


LESSON  VII. 

19.  Find  the  least  common  multiple  of  12,  14, 
and  16.  Am.  96. 

20.  Find  the  least  common  multiple  of  14,   16, 
and  18.  Ana.  1008. 

21.  Find  the  least  common  multiple  of  16,  18, 
and  20.  Ans.  720. 

22.  Find  the  least  common  multiple  of  2,  3,  4,  5, 
and  6.  Ans.  60. 

23.  Find  the  least  common  multiple  of  3,  4,  5,  6, 

7,  and  8.  Ans.  840. 

24.  Find  the  least  common  multiple  of  4,  5,  6,  7, 

8,  and  9.  Ans.  2620. 

25.  Find  the  least  common  multiple  of  5,  6,  7,  8, 

9,  and  10.  Ans.  2520. 

26.  Find  the  least  common  multiple  of  6,  7,  8,  9, 

10,  and  14.  Ans.  2520. 

27.  Find  the  least  common  multiple  of  7,  8,  9,  12, 
14,  and  16.  Ans.  2016. 

28.  Find  the  least  common  multiple  of  8,  9,  12, 
14,  16,  and  18.  Ans.  6048. 

29.  Find  the  least  common  multiple  of  12,  14, 
and  16.  Ans.  336. 

30.  Find  the  least  common  multiple  of  14, 16,  18, 
and  20.  Ans.  5040. 

31.  Find  the  least  common  multiple  of  16,  18, 
and  20.  Ans.  720. 

32.  Find  the  least  common  multiple  of  1,  2,  4,  6, 
10,  and  12.  Ans.  360. 


MISCELLANEOUS   EXAMPLES.  167 

33.  Find  the  least  common  multiple  of  5,  7,  9, 
and  11.  Ane.  3465. 

34.  Find  the  least  common  multiple  of  7,  11,  9, 
and  13.  Am.  9009. 

35.  Find  the  least  common  multiple  of  11,  9,  13, 
and  17.  Am.  21879. 


EXAMPLES  ON  THE  PRECEDING  RULES. 
LESSON  VIII. 

1O1.  1.  The  sum  of  two  numbers  is  12,  and  their 
difference  is  2.     What  are  the  numbers. 

1st.    Add  the  difference  to  the  sum  and  divide 
by  2.     This  gives  the  greater  number. 

2d.  Subtract  the  difference  from  the   sum,  and 
divide  by  2.     This  gives  the  smaller  number. 

Ana.  5  and  7. 
OPERATION. 

12  12 

2  2 

2  )  14  •  2  )_1Q 

7     Greater  number.  5    Less  number. 

192.  NOTE. — Such  sums  are  founded  on  algebraic  problems. 


168  INTEEMEDIATE   ARITHMETIC. 

2.  The  sum  of  two  numlbers  is  125,  and  their 
difference  is  25.     What  are  the  numbers  ? 

Ans.  75  and  50. 

3.  The  sum  of  two  numbers  is  15,  and  their  differ- 
ence is  1.    What  are  the  numbers  ?        Ans.  8  and  7. 

4.  The  sum  of  two  numbers  is  20,  and  their  dif- 
ference is  12.     What  are  the  numbers  ? 

Ans.  16  and  4. 

5.  The  sum  of  two  numbers  is  25,  and  their  dif- 
ference is  11.     What  are  the  numbers? 

Ans.  18  and  7. 

6.  The  sum  of  two  numbers  is  30,  and  their  dif- 
ference is  20.     What  are  the  numbers  ? 

Ans.  25  and  5. 

7.  If  a  certain  number  is  multiplied  by  5,  and  the 
product  divided  by  4,  the  quotient  will  be  25.   What 
is  the  number?  Ans.  20. 

8.  If  a  certain  number  is  multiplied  by  4,  and  the 
product  divided  by  5,  the  quotient  will  be  3*.    What 
is  the  number  ?  Ans.  4. 

9.  If  a  certain  number  is  multiplied  by  7,  and  the 
product  divided  by  2,  the  quotient  will  be  70.   What 
is  the  number?  Ans.  20. 

10.  If  a  certain  number*is  divided  by  4,  and  the 
the  quotient  is  then  multiplied  by  2,  the  product  in- 
creased by  20,  and  the  sum  diminished  by  10,  the 
result  will  be  20.     What  is  the  number  ?        Ans.  20. 


MISCELLANEOUS    EXAMPLES.  169 

11.  A  can  dig  2  rods  per  day,  B  can  dig  4  rods 
per  day,  and  C  can  dig  8  rods  per  day.     What  is  the 
least  number  of  rods  that  will  make  a  number  of  full 
days'  work  for  each  of  the  three  men?  Ans.  8. 

12.  A  gentleman  has  250  gallons  of  sherry,  200 
gallons  of  Madeira,  and  210  gallons  of  claret  wine, 
and  he  desires  to  fill  a  number  of  casks  of  equal  size, 
without  mixing  the  wine.     How  many  gallons  must 
each  cask  hold  ?  Ans.  10. 

13.  Sam.  Brown  has  24  half-dollars,  48  dollars, 
and   200   quarter-dollars.     He   wishes   to   place   an 
equal  number  of  each  in  several  drawers,  not  mixing 
them  or  leaving  any  out.     How  many  will  each 
drawer  contain,  and  what  is  the  least   number  of 
drawers  that  will  answer  the  purpose? 

First  Ans.  8  pieces  in  each  drawer. 

14.  What  is  the  least  number  which  will  divide 
1,  2,  3,  4,  5,  6,  7,  8,  and  9  without  a  remainder? 

Ans. 

15.  A,  B,  and  C  start  together,  and  travel  in  the 
same  direction  around  an  island  1000  miles  in  circum- 
ference.    A  travels  8  miles  a  day,  B  travels  16  miles, 
and  C  travels  32  miles.     What  is  the  least  number 
of  hours  that  will  bring  them  together  ?     How  many 
times  around  the  island  will  each  one  have  traveled  ? 

Last  Ans.   A,  1 ;  B,  2  ;  C,  4. 

15 


170 


INTERMEDIATE   ARITHMETIC. 


CANCELLATION. 
LESSON  IX. 

EXAMPLES. 

193.  1.  Divide  100  by  25. 


2  )  100 
2  )  50 
5  )  25 


OPERATION. 

5  )  25 

5  )_5 

1 


2  x2 


Find  the  prime  factors  of  the  dividend,  and  place 
them  above  the  line.  Find  the  prime  factors  of  the 
divisor,  and  place  them  below  the  line.  Cancel  those 
which  are  common  to  both.  The  product  of  the 
remaining  numbers  is  the  quotient.  Ans.  4. 


2.  Divide  200  by  100. 

OPERATION. 
2  )   200  #X$X2X$X 

2  )  100  #  X  %  X  $  X 

2~7~50 
5  )~25 


2  )  100 

2  )  50 

5  )~25 

*) 


1 


Ans.  2. 


QAN  CELLAT1  ON.  1  V  1 


3.  Divide  24  by  8.  Am.  3. 

4.  Divide  48  by  16.  Ans.  3. 

5.  Divide  90  by  32.  Am.  3. 

6.  Divide  102  by  64.  Am.  5. 

7.  Divide  384  by  128.  Am.  3. 

8.  Divide  768  by  256.  Am.  3. 

9.  Divide  1536  by  128.  Ans.  12. 

10.  Divide  2072  by  24.  Am. 

11.  Divide  4144  by  48.  Ans. 

12.  Divide  8288  by  96.  Am. 

13.  Divide  16576  by  192.  Am. 

14.  Divide  33152  by  384.  Am. 

15.  Divide  66304  by  768.  Ans. 

16.  Divide  132608  by  1536.  Am. 

17.  Divide  265216  by  3072.  Am. 

18.  Divide  530432  by  1536.  Ans. 

19.  Divide  1080864  by  3072.  Ans. 

20.  Divide  2121728  by  1536.  Ans. 

21.  Divide  200000  by  2000.  Am.  100. 

22.  Divide  40400  by  400.           ,  Am.  11. 

23.  Divide  50000  by  5000.  Am.  10. 

24.  Divide  50500  by  500.  Am.  11. 

25.  Divide  60000  by  6000.  Am.  10. 

26.  Divide  70000  by  7000.  Am.  10. 

27.  Divide  4243556  by  1536.  Ans. 

28.  Divide  90000  by  9000.  Am.  10. 

29.  Divide  8487112  by  3072.  Ans. 


172 


INTERMEDIATE   AKITHMETIC. 


CHAPTER  X. 

F  Pt  A  C  T  I  0  ff  S . 

LESSOR  I. 

194.  THE  unit  1  means  one  entire  object,  or  a 
whole  object. 

195.  If  that  object  is  cut  into  two  equal  parts, 
each  part  is  called  one  half  of  the  object,  and  is  thus 
written,  -J,  the  figure  1  above  the  2,  with  a  line  drawn 
between  them. 

196.  If  the  object  is  divided  into  3  equal  parts, 
each  part  is  called  Q)  one  third. 

197.  If  the  object  is  divided  into  four  equal 
parts,  each  part  is  called  (|)  one  fourth,  etc.,  etc. 


-J-ifi 

read  one  half. 

T1! 

is  read  one  eleventh. 

4 

"       one  third. 

iV 

"       one  twelfth. 

i 

"       one  fourth. 

TV 

"       one  thirteenth. 

t 

"       one  fifth. 

A 

"       one  fourteenth. 

i 

"       one  sixth. 

iV 

"       one  fifteenth. 

i 

"       one  seventh. 

iV 

"       one  sixteenth. 

t 

"       one  eighth. 

iV 

"       one  seventeenth. 

^ 

"       one  ninth. 

iV 

"       one  eighteenth. 

TV 

"       one  tenth. 

TV 

"        one  nineteenth. 

198.  A  fraction  is  an  equal  part  of  a  unit.  The 
number  below  the  line  is  called  the  denominator, 
and  shows  into  how  many  parts  the  object  is  divided. 


FRACTIONS.  173 

199.  The  number  above  the  line  is  called  the 
numerator,  and  shows  how  many  of  the  equal  parts 
of  the  fraction  are  used. 

200.  In  J,  the  figure  4  is  the  denominator,  and 
the  3  is  the  numerator.     It  shows  that  the  object  was 
divided  into  4  equal  parts,  of  which  3  parts  were  used. 

201.  A  fraction  is  less  than  a  whole  one. 
There  are  several  kinds  of  fractions  : 

202.  A  proper  fraction  is  one  whose  numerator 
is  less  than  the  denominator ;  thus,  £. 

SOJ5.  An  improper  fraction  is  one  whose  numer- 
ator is  equal  to  or  greater  than  the  denominator; 
thus,  f  or  -2-. 

204.  A  compound  fraction  is  a  fraction  of  a 
fraction,  or  a  fraction  connected  to  another  by  the 
signo/;  thus,  |  off 

205.  A  mixed  fraction  is  a  fraction  and  a  whole 
number ;  thus,  2 1-,  and  is  read,  two  and  one  half. 

200.  A  complex  fraction  is  one  which  has  a 
fraction  for  the  numerator  and  a  fraction  for  the  de- 
nominator, or  a  fraction  in  either  numerator  or 

denominator ;    thus   —   is    read,   one   half  divided 

by  three  fourths ;  or  —  is  read,  one  divided  by  one 

half. 

2O7.  The  value  of  a  fraction  is  the  quotient 
found  by  dividing  the  numerator  by  the  denomina- 
tor. The  value  of  a  proper  fraction  must  be  less 
than  1.  The  value  of  an  improper  fraction  must  be 
equal  to  or  greater  than  1. 
15* 


174  INTERMEDIATE   ARITHMETIC. 


LESSON  II. 

PROPOSITIONS. 

2O  8.  If  the  numerator  and  denominator  of  a 
fraction  are  divided  by  the  same  number,  the  value 
of  the  fraction  is  not  changed;  thus  T8¥,  divide 
both  numerator  and  denominator  by  2,  and  the  frac- 
tion becomes  . 


If  the  numerator  and  denominator  of  a 
fraction  are  multiplied  by  the  same  number,  the 
value  of  the  fraction  is  not  changed  ;  thus  ^,  mul- 
tiply both  numerator  and  denominator  by  2,  and  the 
fraction  becomes  -£$. 

21O.  By  multiplying  the  denominator  of  a  frac- 
tion, the  value  of  the  fraction  is  decreased. 

311.  By  dividing  the  numerator  of  a  fraction, 
the  value  of  the  fraction  is  decreased. 


LESSON  III. 

SI  2.  To  reduce  a  fraction  to  its  lowest  terms. 

A  fraction  is  reduced  to  its  lowest  terms  when 
the  numerator  and  denominator  are  prime  to  each 
other. 

213.  The  terms  of  a  fraction  are  the  numerator 
and  denominator. 


REDUCTION   OF  FRACTIONS.  175 

.  To  reduce  fractions  : 

EULE. 

Divide  the  numerator  and  denominator  by  their 
greatest  common  divisor. 

EXAMPLES. 
1.  Reduce  T%  to  its  lowest  terms. 

OPERATION. 

3  is  the  common  divisor.     Divide  the  numerator 
and  denominator  by  3. 

3  )  A  =  !,  Ans. 

2.  Reduce  -f%,  -fc,  -ff  to  their  lowest  terms. 

Ans.  |,  etc. 

3.  Reduce  -££  ,  ^f,  TW  to  their  lowest  terms. 

Ans.  -f,  etc. 

4.  Reduce  f,  f,  T8g-,  -J-|  to  their  lowest  terms. 

Ans.  ^,  etc. 

5.  Reduce  T5¥,  ff,  ^-|,  4^  to  their  lowest  terms. 

Ans.  |-,  etc. 

6.  Reduce  ^f  ,  fg-,  |f  ,  -ffy  to  their  lowest  terms. 

Ans.  |,  etc. 

7.  Reduce  if,  f|,  -^-,  |f  to  their  lowest  terms. 

etc. 


8.  Reduce  Jft,   ff,   f|,   fJ-J   to    their    lowest 
terms.  Ans.  %,  etc. 


176  IXTEEMEDIATE   AKITHMETIC. 


LESSON  IV. 

0.  Reduce 

fo'lnf  to  *ts  lowest  terms. 

OPERATION. 

Reduce  it  to  its  prime  factors  : 

Numerator. 

Denominator. 

2  )  2000 

#  X  #  X  #  X  2  X  0  X  $  X  0 

2  )  3000 

2  )  1000 

#X#X#X0X0X$X3 

2  )  1500 

2  )  500 

2  )  750 

2  )  250 

2 

5  )  375 

5  )  125 

3 

5  )  75 

5  )  25 

5  )  15 

5  )  5 

3  )  3 

1 

1 

10.  Reduce  yVA  *°  its  lowest  terms.          Ans.  f. 

11.  Reduce  yVo5^  to  its  lowest  terms.      Ans. -f^. 

12.  Reduce  £J{-  to  its  lowest  terms. 

13.  Reduce  -f-fc  to  its  lowest  terms. 

14.  Reduce  £{j»{j-  to  its  16*  west  terms. 

15.  Reduce  |--{j-{j-  to  its  lowest  terms.  .4ns.  f. 

16.  Reduce  f££  to  its  lowest  terms. 

17.  Reduce  -g-*-g-  to  its  lowest  terms. 

18.  Reduce  -J- $-{}{}•  to  its  lowest  terms. 

19.  Reduce  ||^  to  its  lowest  terms. 

20.  Reduce  MM  to  its  lowest  terms.      Ans. 


21.  Reduce  J-§-g--g-  to  its  lowest  terms. 

22.  Reduce  £-§-${{•  to  its  lowest  terms. 


REDUCTION   OP  FRACTIONS.  177 

23.  Reduce  -i  •){>(}  to  its  lowest  terms.  Ans.  ?.;;. 

24.  Reduce  ^VoVo- to  ^ts  lowest  terms.  -4>zs.  £||. 

25.  Reduce  ^VuV  to  its  lowest  terms.  Ans.  ^T. 
20.  Reduce  -HISJj-  to  its  lowest  terms.  Ans.  ff 

27.  Reduce  JJ-ooo- to  its  lowest  terms.      ^l;z*.  H<. 

28.  Reduce  -J-Jg-JJ"  to  its  lowest  terms.      Ans.  f£. 

29.  Reduce  {$-%$$  to  its  lowest  terms.      Ans.  ^ 

30.  Reduce  |-jj-f {)-{)-  to  its  lowest  terms.    Ans.  J-J*. 

31.  Reduce  -ffffl} to  its  lowest  terms.    ^4ws.  -}-||. 

32.  Reduce  fflHHjto  its  lowest  terms.  Ans.  \H\. 

33.  Heduce  -£-}•$$  to  its  lowest  terms.     Ans.  -}|il  J. 

34.  Reduce  iJ^-^-J  to  its  lowest  terms.    -4?zs. 

35.  Reduce  Jf||  to  its  lowest  terms.    Ans, 


LESSON  Y. 

.  To  reduce  fractious  to  fractions  having  a 
proposed  denominator. 

KULE. 

I.  Heduce  the  fraction  to  its  lowest  terms. 

II.  Divide  the  given  denominator  by  the  denomi- 
nator of  the  fraction  reduced. 

III.  Multiply  the  numerator  and  denominator  of 
the  reduced  fraction  by  the  quotient  arising  from 
dividing  the  reduced  denominator  into  the  proposed 
denominator. 

IV.  The  result  will  be  the  required  fraction. 


178  INTERMEDIATE  ARITHMETIC. 

EXAMPLES. 

1.  Reduce  -f-  to  a  fraction  whose  denominator  will 
be  12. 

OPEKATIOX. 

Reducing  $  to  its  lowest  terras  gives  us  f  .  Di- 
viding the  proposed  denominator,  12,  by  the  denom- 
inator of  the  reduced  fraction,  3,  the  quotient  is  4. 
Multiplying  both  numerator  and  denominator  of  the 
reduced  fraction  -|  by  4,  we  get  T8^  for  the  required 
fraction. 

210.  NOTE.  —  The  above  is  the  first  step  towards  addition  and 
subtraction  of  fractions,  and  should  be  carefully  considered. 

2.  Reduce  ^,  f,  f,  T6g-,  |-|  to  fractions  whose  de- 
nominators will  be  60.  Ana.  |«,  |f,  |1,  f&  |«, 

3.  Reduce  ^,  f  ,  f,  T9^  to  fractions  whose  denom- 
inators will  be  36.  Ans.  4$,  J  |,  f  f  ,  |f 

4.  Reduce  f  ,  |,  y^,  -}-|  to  fractions  whose  denom- 
inators will  be  180.  4*M,Mvtt!»*tt>itf 

5.  Reduce  f  ^,  -^  to  fractions  whose  denomina- 
tors are  30.  Ans.  J$,  g6^ 

6.  Reduce  f  ,  £  ,  £J  to  fractions  whose  denomina- 
tors shall  be  21.  ^s.  ^_,  -if,  |]. 

Reduce  ^,  -|-,  f  ,  ^  to  fractions  whose  denominators 
will  be  120.  Ans.  jfr,  T^,  ffr,  ^. 

8.  Reduce  |-,  -J-?  J,  ^  ^  to  fractions  whose  de- 
nominators will  be  60.  Ans.  -jft,  |f,  f?,  fj-,  J-g. 

9.  Reduce  |,  f,  |,  -|,  f  to  iractions  wThose  de- 


nominators will  be  60.  Ans.  f£,  ££,  f-J,  ^,  |g. 

10.  Reduce  f  ,  f  ,  |,  T\,  -J-f  to  fractions  whose  de- 

nominators will  be  60.  Ans.  fg-,  f|,  |$,  fj,  fj. 


REDUCTION    OF   FRACTIONS.  179 

11.  Reduce  4=£  to  a  fraction  whose  denominator 
will  be  12.  Ans.  \*>. 

12.  Reduce  2,    3,  4,  5,  C,  7,    8,  9  to  a   fraction 
whose  denominator  will  be  5. 

Ans.  V,  V,  *<?,¥, -V>  :VVV- 

13.  Reduce    3,    4,    5,  6,  7,  8,  9,  10  to   fractions 
whose  denominators  will  be  8.     Ans.  %£,  *£,  V>  etc. 

14.  Reduce  12,  13,  14,  15,  16  to  fractions  whose 
denominators  will  be  10.         Ans.  iffi,  ^,  Yu°->  ctc- 


LESSON  VI. 

317.  To  reduce  a  mixed  number  to  an  improper 
fraction. 

RULE. 

I.  Jteduce  the  fractional  part  to  its  lowest  terms. 

II.  Multiply  the,   whole   number  by   the  reduced 
denominator,   and   to    the   result   add   the    reduced 
numerator. 

III.  Write  the  sum  over  the  reduced  denominator. 

EXAMPLES. 

1.  Reduce  12-&  to  an  improper  fraction. 

OPERATION. 

First  reduce  -fa  to  its  lowest  terms,  which  is  £. 
Multiply  the  whole  number,  12,  by  the  reduced 
denominator,  2,  making  24,  and  add  the  reduced 
numerator,  1,  making  25.  Write  25  for  the  new 
numerator,  and  the  reduced  denominator,  2,  for  the 
new  denominator.  The  fraction  required  will  be  ^-. 


180  INTERMEDIATE   ARITHMETIC. 

2.  Reduce  2TSF,  6f,  9|,  to  improper  fractions. 

Am.  |,  -y->  ¥• 

3.  Reduce   3^-,   4-^V,    19ff,  24fV  to   improper 
fractions.  J^s.  JJ-,  y,  4£,  4f. 

4.  Reduce   11J-|,  21f{{,  84^,  to  improper  frac- 
tions. ./I  /is.  Af,  -6/,  i-p, 

5.  Reduce  7JJ,  6|f,  2ff,  9|J|,  to  improper  frac- 
tions. Am.  -VS  %  V?  ¥• 

6.  Reduce  4ff,  25ft,  61Tr'/T,  to  improper  frac- 
tions. Ans.  -2/,  J^SL,  iJA. 

7.  Reduce  5-1-,  6J,  7J,  8-},  9J,  to  improper  frac- 

tions. ^.  v-,  ¥,  ¥,  ¥,  V- 

8.  Reduce  10^-,  11J,  12|^,  13^,  to  improper  frac- 
tions. -4^.  v,  ¥,  ^F,  W-- 

9.  Reduce  14-^0,   15T1?,  16^,  to  improper  frac- 
tions. Ans.  JTy-,  ij^6-,  -^-. 

10.  Reduce  214-,  22-J-,  23J,  24|,  to  improper  frac- 
tions. Ans.  *£-,  sg-,  -9T3-,  ifi. 

11.  Reduce  31^-,  32J,  334;,  34|,  to  improper  frac- 
tions. Ans.  $/-,  -V-,  ip,  ip-. 

12.  Reduce  41^,  42|-,  43^,  44f,  to  improper  frac- 
tions. Ans.  i|±,  ip,  ^,  -2-p. 

13.  Reduce  51f,  52|-,  53i,  55  J,  to  improper  frac- 
tions. u4^5.  1|A,  AJJ-,  ^|A,  Afi. 

14.  Reduce  61f,  62^,  63f,  to  improper  fractions. 


15.  Reduce  71|-,   81^,   91J4,  to   improper  frac- 
tions. "  Ans.  ^fi,  -Vo9-,  1ifa- 

16.  Reduce   81f,    82f,    83Trv,   84-^,  to   improper 
fractions.  Ans.  *  -^. 


REDUCTION   OP   FRACTIONS.  181 

17.  Reduce  911,  92J,  93J-,  94J,  9o£,  to  improper 
fractions.  Ans.  J-JP,  *$*,  ^,  H'S  ^ 

18.  Reduce  1^  ^Vo,  3/0\,  4-^,  5^,  to  im- 
proper fractions.  Ans.  -}  •>  »  ,  *  »  »  -,  -»  «  »  ,  *  j|  »  ,  -«  »  ||  . 

19.  Reduce  6^,   7^,  8,7,,  to  improper  frac- 


tions. Ans.***,  ]|Jil,  ?»». 

20.  Reduce  2-jfo,  3-g^,  4^,  5^,  to  improper  frac- 
tions. Ans.  W,  W,  W,  W- 

21.  Reduce  2||},  3|-J,  4||,  to  improper  fractions. 


22.  Reduce  1T2^,  1^,   1^-,  to  improper  frac- 
tions. Ans.  W,  f,  «J. 

23.  Reduce  2£,  3J,  4J-,  5J-,  to  improper  fractions. 

Ans.  1,  J,  |,  -V. 

24.  Reduce  12|,  13J,  14J,  15J,  to  improper  frac- 
tions. ^in«,  *^,  V,-V,  V- 

25.  Reduce  121-J-,  131J,  141£,  to  improper  frac- 
tions. Ans.  *f±,  if*,  ip. 

LESSON  VII. 

SI  8.  To  reduce  improper  fractions  to   mixed 
numbers. 

EULE. 

Divide  the  numerator  by  the  denominator,  and 
write  the  remainder,  if  any,  over  the  denominator. 


EXAMPLES. 

1.  Reduce  %£-  to  a  mixed  number. 

Divide  21  by  4.     The  answer  will 
be  5  and  1  over,  which  is  written,  5£. 
16 


OPERATION. 

4  )  21  (  51- 
20 


182 


INTERMEDIATE    ARITHMETIC. 


.  To  prove  the  sum  correct,  multiply  the 
who!e   number   by  the   denominator,  and   add   the 


numerator. 

2.  Reduce 

3.  Reduce 

4.  Reduce 

5.  Reduce 
G.  Reduce 

7.  Reduce 

8.  Reduce 

9.  Reduce 

10.  Reduce 

11.  Reduce 

12.  'Reduce 

13.  Reduce 

14.  Reduce 

15.  Reduce 

16.  Reduce 
IV.  Reduce 

18.  Reduce 
numbers. 

19.  Reduce 

20.  Reduce 

21.  Reduce 


4x5  =  20,         2  0  -f-  1  =  2  1  ,  or 


-V3-,  -\9,  to  mixed  numbers. 
^/,  -1^,  -3/,  ^-,  to  mixed  numbers. 
3^,  -%4-,  -*-jp,  to  mixed  numbers. 
J^-,  -^°-,  -y-,  *£-,  to  mixed  numbers. 
%£-,  ^np-,  -S^-8-,  to  mixed  numbers. 
*£-,  -1/,  -^-,  ^-,  -5e§-,  to  mixed  numbers. 
V->  Y->  H-J  ¥oS  to  mixed  numbers. 
^9-,  Jy9/,  ^-,  to  mixed  numbers. 
-4g3,  -\7,  -y-,  ^-f1.  to  mixed  numbers. 
-623-,  -^,  i|^,  J-p-,  to  mixed  numbers. 
J-f*-,  J-f^,  -^J-S-,  ^p,  to  mixed  numbers. 
-Mp-,  -3-^J-,  ^-§-^-,  ^f-5-,  to  mixed  numbers. 
-2-J1-,  i-g-i,  -5^-S.j  to  mixed  numbers. 
ifi,  -8TV9-,  i|p,  to  mixed  numbers. 

l-9-,  to  mixed  numbers. 
U-,  to  mixed  numbers. 
f|-|,   f^-,   to  mixed 


i-3- 


fff,  -J-g-g-,  |9f,  to  mixed  numbers. 
VaS  W>  W>  Wj  to  mixed  numbers. 
V98-j  W»  W>  to  mixed  numbers. 


'br  answers,  see  reduction  to  mixed  numbers.] 


ADDITION    OF   FRACTIONS.  183 

ADDITION  OF  FRACTIONS. 
LESSON"  VIII. 

25SO.  When  the  fractions  have  the  same  denom- 
inator. 

KULE. 

Add  the  numerators  and  place  the  sum  over  the 
common  denominator.  If  an  improper  fraction  is 
the  result,  reduce  it  to  a  mixed  number. 

EXAMPLES. 

1.  Add  i,  i,  |,  *,  i. 

OPERATION. 

Adding  the  numerators  gives  5,  which  written 
over  the  denominator  gives  |,  which  reduced  is  equal 
to  2. 


2.  Add  I,  i  -f,  J.  .4w*.  4  or  If 

3.  Addf,f,},J.  ^^5.  for  21 

4.  Add  f,  -£,  J,  f  .  ^;is.  Y-  or  2f  or  2J- 

5.  Add  |,  -|,  -I,  4.  ^t?zs.  \°  or  If. 

6.  Add  i,  I,  },  |-  ^n5.  ^  or  2  J  or  2f 

7.  AddJ,  J,  {-,  f  ^w.  Vorlf. 

8.  Add  TV,  fV,  TV,  TV  Am.  f  g-  or  2. 

9.  Add  TV,  T2T,  f\,  f?-  -4n*  |f  or  1T6T. 

10.  Add  A,  T7^  fV3  TV  ^»*  H  or  1TV 

11.  Add  TV,  T5F,  T4j,  TV  ^n5.  fl  or  1TV 

12.  Add  TV,  T5¥,  TV,  H-  •4w«-  ff  or  Iff  or  If 

13.  Add  ^,  A)  A.  TV  -4™.  ?i  or  1  A  or  if 


184 


INTERMEDIATE    ARITHMETIC. 


14.  Add  T5c,  TV,  T9c>  tt- 

15.  Add  W,  T'T,  14,  -H- 

16.  Add  A,  T\>  H>  VI- 

17.  Add  T\,  A,  f«,  f', 

18.  Add  ¥V,  A»  A)  W- 

19.  Add  ^  /T,  /T,  ¥«T. 

20.  Add  A,  A,  A,  it- 

21.  Add  if,  Jj,  li,  !i- 


or 


If  or  2- 
or  2T\. 

fl  or  2. 
f|  or  2. 

or  !- 


or  ijf  or  1T\. 


or 


LESSON  IX. 

.  When  the  denominators  are  prime  to  each 
other. 

EULE. 

Multiply  the  numerators  by  all  the  denominators 
except  its  own  for  a  new  numerator. 

Multiply  all  the  denominators  together  for  a  new 
denominator. 

Add  the  new  numerators  and  write  their  sum  over 
the  common  denominator. 


1.  Add  J  and  |. 


EXAMPLES. 


OPERATION. 


JL  JL  JL  =  (5X1)  i  (4><1)  _  5+4  =  A 

4    "    5  20  20  20          20 


2.  Add 

3.  Add 

4.  Add 

5.  Add 


or 

or  iff. 
Ans.  f|£  or  l^ft.  or  1ft. 
or 


ADDITION   OF   FKACTIO^S.  185 


6.  Add  Jr-HH-T7r- 

7.  AddJ--H-ff,  An*. 

8.  AddA+t+A-  Ans.  *!;•'. 

9.  Addi+iV+H-  -4n*.  rtff 

10.  Add  TV+TV-H.  Ans.  -«  »  J. 

11.  Add 

12.  Add 


14.  Add 

15.  Add 

16.  Add 

17.  Add 

18.  Add 

1  9.  Add 

20.  Add  ^+,ir+^.  Ans.  fj  Hi;?,. 

21.  Add 


LESSOR  X. 

.  When  the  denominators  are  not  all  prime 
to  each  other. 

RULE. 

I.  Find  the  least  common  multiple  of  the  denom- 
inators. 

II.  Divide  the  least  common  multiple  by  the  de- 
nominator of  every  one  of  the  fractions. 

III.  Multiply  the  numerator  of  every  one  of  the 
fractions  by  the  quotient  found  by  dividing  the  least 
common  multiple  by  the  denominator.      The  result 
will  be  the  numerator  of  the  fraction. 

16* 


186  INTERMEDIATE  ARITHMETIC. 

IV.  Add  the  several  numerators  thus  found,  and 
write  the  sum  over  the  least  common  multiple. 

V.  If  the  sum  makes  an  improper  fraction,  reduce 
it  to  a  mixed  number,  and  reduce  the  fractional  part 
to  its  lowest  terms  if  possible. 

EXAMPLES. 

1.  Add  i+f +£+£. 

OPERATION. 

The  common  multiple  of  2,  3,  4  and  G  is  12.  Since 
the  multiple  or  denominator  will  be  the  same  for 
them  all,  it  would  do  as  well  to  write  it  under  one 
line  of  sufficient  length  to  hold  all  of  the  new  numer- 
ators ;  thus : 

6+8  +  9+10 
12 

Divide  the  denominator  2  into  the  multiple  :  2  in  12, 
G  times.  Multiply  the  numerator  1  by  6  :  6  times  1 
are  6.  Write  the  figure  6  on  the  line  above  the  mul- 
tiple, as  above.  Take  the  next  denominator :  3  in 
12,  4  times.  4  times  2  are  8.  Write  the  figure  8 
above  the  line.  Take  the  next  denominator:  4  in 
12,  3  times.  3  times  3  are  9.  Write  the  figure  9  on 
the  line.  Take  the  next  denominator:  6  in  12,  2 
times.  2  times  5  are  10.  Write  the  figure  10  above 
the  line.  The  pupil  must  not  omit  to  \yrite  the  sign 
(  +  )  plus  between  the  numerators.  Add  the  several 
numerators : 

6+8+9+10=33 


ADDITION   OF   FRACTIONS.  187 

The  answer  is  33  for  the  numerator  and  12  for  the 
denominator,  or  y,  which  is  an  improper  fraction. 
Reduce  the  improper  fraction  to  a  mixed  number  l>y 
dividing  the  numerator  by  the  denominator:  12  in 
33,  2  times  and  9  over.  The  answer  as  found  is  (-J  ,",) 
2  whole  ones  and  (fy)  nine  twelfths.  Reducing  ^ 
to  its  lowest  terms  by  dividing  both  numerator  and 
denominator,  9  and  12,  by  3,  we  get  3  and  4.  3  for 
the  numerator  of  the  reduced  fraction,  and  4  for  its 
denominator.  The  true  answer  is  2|. 

223.  NOTE. — All  answers  should  be  reduced  to  their  lowest 
terms. 

224,  NOTE. — By  making  the  pupil  analyse  a  sum  as  given 
above.  6*11  the  preceding  rules  will  be  repeated,  and  the  pupil  will 
begin  to  understand  the  meaning  of  what  he  has  learned  in  what  is 
called  "  Properties  of  Numbers." 

2.  Add  J+f  Am.  |, 

3.  Add  i-f  J.  Ans.  TV 

4.  Add  i-ff  Ans.  /~, 

5.  Add  £+f  Ans.  Jf 

6.  Add  J-+ f  Ans.  }$. 

7.  Add  -i-ff  Ans.  -J-J-. 

8.  Add  £-1-1. 

9.  Add|+TV 

10.  Add 

11.  Add 

12.  Add 
13. 

14.  Add 

15.  Add 


188 


INTERMEDIATE   ARITHMETIC. 


16.  Add  i 

17.  Add 

18.  Add  i*T-f  TV 

19.  Add 

20.  Add 

21.  Add  i+T28-f  sV 

22.  Add 

23.  Add 

24.  Add 

25.  Add 

26.  Add 


Ans. 


Ans.  -f 
•.  14. 


.   I 


Ans.  If. 


z? 

28.  Add 

29.  Add  TV+ M + M+T^s. 

30.  Add 

31.  Add 

32.  Add 
33. 


Ans.  1|-. 


Ans.  2-J. 


34.  Add  J  + !  +  -?-. 


35. 
36. 

37. 

38.  Addf+f  +  f-ff 

40.  Add  |+i-f  t-f  i-hf 


Ans. 


41. 

42.  Add 


Ans. 


ADDITION    OF   FRACTIONS.  189 

43.  Add  l+4+3+5+ft+.j  1-  Ans.  52%3Tr 

44.  Add  £+A+5+£  +  T7-  +  T3f. 

45.  Add  i+l+£+|+JV+  jf 

4G.  Add  1+4+1 

47.  Add 


48. 

-  Httttt- 

49.  Add  -H^-|H-IV+12f-f-i4-f 

50.  Add 

51.  Add 

52.  Add 

6HHK 

53.  Add^+|+|+f 

54.  Add  f  +TV+f|+|f. 

55.  Add|+TV+i|. 

56.  Add  i+i+J+|-f.|. 

57.  Add  J-+J-+I+I+-1-. 

58.  Add  l+i+i+i+i. 

59.  Add  i+i+i+i+TV  Ans. 

60.  Add  1+1+1+^+^.  Ans. 

61.  Add  i+J-f-f. 

62.  Add  l+f+i 

63.  Add  |+^+|+f  ^^5.  2f 

64.  Add     J+l+l+4+l+l+l+l+.-V+A+TV 

Ans. 


190  INTERMEDIATE   ARITHMETIC. 

LESSOR    XL 

225  •  To  add  mixed  numbers. 

RULE  I. 

I.  Add  the  whole  numbers. 
II.  Add  the  fractional  parts. 
III.  Add  the  sum  of  the  fractional  parts  to  the 
sum  of  the  whole  numbers  for  the  ansioer. 

226.  Or: 

EULE  II. 

JReduce  the  mixed  numbers  to  improper  fractions, 

and  add  the  improper  fractions  thus  formed,  as  in 
addition  of  fractions. 

1.  Add  21  +  21,  Ans.  4f 

2.  Add  31  +  31.  Ans.  6T%. 

3.  Add  4i  +  4|.  Ans.  8¥V 

4.  Add5|-  +  5i.  Ans.  IQii. 

5 .  Add  61  +  61  +  71  Ans.  1 9^. 

6.  Add  7f +  7f +  7f  Ans.  22f. 

7.  Add  8i  +  8J-  +  8TV-  ^1«*.  24f-f j}, 

8.  Add  Sf  +  8f  +  8fV  -4w&  25|ff . 

9.  Add  9|  +  9|  +  9|-  +  9f  Ans.  39^. 

10.  Add  10i  +  10f +  10T\.  Ans.  31  ^W 

11.  Add  lli  +  llf +  11|.  -4ws.  34|. 

12.  Add  12f +  13i  +  13f +  181  Ans. 

13.  Add  13f +  14|  +  21TV  +  22TV  Ans.  7 

14.  Add  14f +15|+19TV  +  12TV  Ans. 

15.  Add  15i+16i+18i  +  19TV  ^.ws.  63||f 

16.  Add  16f+17fV  +  lST\  +  19fi.  Ans.  71 1|. 

17.  Add  17f  +  18-J-+ 19^  +  24^-.  Ans.  80|f 


ADDITION    OF   FRACTIONS.  191 

18.  Add  18J-+194  +  20J  +  21J  +  22-}-.   Ans.  101§.J, 

19.  Add  1 


20.  Add  20f  +  30J  +  40£  + 


LESSOX  XII. 

21.  A  grocer  bought  4  firkins  of  butter,  weigh- 
ing respectively  24J-,  25  J-,  26},  27  j  pounds.     How 
many  pounds  in  the  4  firkins?  Ans.  103fJ. 

22.  A  person  traveled  in  one  day  25  J-  miles,  26  f 
in  the  next,  24*  in  the  third,  and  27-J-  in  the  fourth. 
How  many  'miles  did  he  travel  in  4  days  ? 

Ans.  103JJ. 

23.  Jas.  Brown  owes  2|  dollars  to  one  man,  6$ 
dollars  to  a  second,  and  24^  dollars  to  a  third.    How 
much  does  he  owe  to  the  three  men?        Ans.  $33^-. 

24.  A  merchant  sold  6  hams.     The  weight  of 
them  was  as  follows:  10J,   12J,  13J,  9f,   15f,   10?- 
pounds.     How  many  pounds  did  they  weigh? 

Ans.  77-J. 

25.  An  auctioneer  sold  4  farms,  containing  the 
following  number  of  acres:    17|,   25£,   46T\,   64^ 
acres.    How  many  acres  were  in   the  four  farms 
together;  and  how  much  did  they  cost,  at  one  dol- 
lar an  acre  ?  Ans. 

26.  A  man  sold  three  horses  for  the  folio  win  o- 

O 

sums  of  money:  the  first  horse  for  22-J-  dollars,  the 
second  horse  for  30-J-  dollars,  and  the  third  horse  for 
25^-  dollars.  How  much  did  he  get  for  the  three 
horses  ?  Ans. 


192  INTERMEDIATE   ARITHMETIC. 


LESSON  XIII. 

227.  When  whole  numbers  and  mixed  numbers 
are  found  in  the  same  problem. 

RULE  I. 

Add  the  ichole  numbers  to  those  of  the  mixed 
numbers. 

Add  the  fractions  of  the  mixed  numbers. 

Add  the  sum  of  the  fractions  of  the  mixed  num- 
bers to  the  sum  of  the  whole  numbers.  The  sum  will 
be  the  answer  required. 

228.  Or: 

RULE  II. 

Consider  the  denominator  of  the  whole  number  as 
1.  Reduce  the  mixed  numbers  to  improper  fractions, 
and  add. 

229.  NOTE. — Since  the  denominator  of  the  whole  number  is  1, 
multiply  the  numerator  of  the  whole  number  (which  is  itself  the 
whole  number)  by  the  common  multiple  for  the  new  denominator. 

EXAMPLES. 

1.  Add  2,  3|,  4£. 

OPERATION. 

Consider  the  denominator  of  the  whole  number  2 
as  1,  and  reduce  the  mixed  numbers  to  improper 
fractions.  We  then  have  f+|-f  V,  tne  common 
denominator  of  which  is  4. 


ADDITION    OF   FRACTIONS.  193 

Write  the  4  beneath  the  line,  as  below : 

8  +  14  +  17        39 

-  =  —  =  9?     Ans. 
4  4 

Begin  the  division  of  the  multiple  by  the  denom- 
inators of  the  given  fractions,  and  multiply  the 
denominators;  thus:  1  in  4,  4  times;  4  times  2  are 
8,  the  first  numerator.  2  into  4,  2  times ;  2  times  7 
are  14,  the  second  numerator.  4  in  4,  1  time;  once 
17  is  17,  the  last  numerator.  Add  the  numerators, 
8  +  14+17  =  39,  and  write  it  above  the  denominator, 
4 ;  as,  y-.  Reducing  \°-  to  a  mixed  number  gives, 
4  into  39,  9  times  and  J  over.  The  answer  will 
be  9}. 

2.  Add  2  +  2-J-  +  2f .  Ans.  7f 

3.  Add  3+4  +  2J.  Ans.  9J. 

4.  Add  4  +  5}  +  6f  Ans.  1 5TV 

5.  Add  5-f +  5  +  5f  Ans.  15f|, 

6.  Add  6f +  4  +  SJ +20.  Ans.  38f 

7.  Add  7f  +  7f  +  7  J  +  4.  ^4m.  2 7-1%  • 

8.  Add  8|  +  8f  +18JJ-  +  19.  ^ws.  55fJ. 

9.  Add  9J-  +  10f  +11-J +  14.  Ans.  45  f. 

10.  Add  lO^+llf+21  +  22.  ^W5.  64-JJ. 

11.  Add  llf +  5^+18  +  22.  Ans.  57*. 

12.  Add  12i  +  13T5¥+lC  +  80.  Ans.  121TV 

13.  Add  13-|  +  14|  +  18TV  +  14.  ^4/is.  60-J-f 

14.  Add  14f +  18  +  81 +  9/T.  Ans.  123^. 

15.  Add  22-J  +  19J  +  14J +  18.  Ans.  74|. 

16.  Add  1411  +  248^+188.  Ans.  577f|, 

17.  Add  2263j  +  1497f +  100.  Ans.  380-1. 

18.  Add  1999TV  +  876f|  +  10  +  12.          Ans.  2898. 

17 


194 


INTERMEDIATE    ARITHMETIC. 


SUBTRACTION  OF  FRACTIONS. 

LESSON  XIV. 

230.  SUBTRACTION  OF  FRACTIONS  is  the  process 
of  subtracting  one  fraction  from  another. 

CASE   I. 

When  the  fractions  have  the  same  denominator. 

RULE. 

231.  Subtract  one  numerator  from  the  other,  and 
write  the  difference  over  the  common  denominator. 


EXAMPLES. 


1.  From  -|  take 


Ans. 


OPEEATION. 

The  denominators  being  the  same  and 
common  to  both,  subtract  the  numerator  3  from  the 
numerator  4,  and  write  the  difference,  1,  over  the 
common  denominator,  5.  The  fraction  thus  formed 
is  . 


2.  From  -|  take  f . 

3.  From  T9F  take  T5¥. 

4.  From  \\  take  \\. 

5.  From  \\  take  -J-J. 

6.  From  §-f  take  f », 


7.  From  -|*u 

8.  From  jj~£-J  take 

9.  From  H\  take 


^4ns.  -|. 

Ans.  ^9. 

^4^5.  -^j. 

^1^5.  -33¥  or  Jg-. 

^.^s,  -§?¥. 

Ans.  J-g-g-  or  -|. 

JLn5.  -J-g-}. 

^4??5.  -g-9-^. 


sriJTKACTJOX    OF    FKACTIO.NS.  195 


10.  From  -ffa\  take  T1/'.7,.  Ans. 

11.  From  -«  ||  take  |{f.  Ans. 

12.  FtomifftakeiH-  Ans- 

13.  From  mi  take  fffo.  Ans. 

14.  From           take          .  .4*w. 


LESSON  XV. 

CASE   II. 

.  When  there  are  two  fractions  whose  nu- 
merators are  1,  and  whose  denominators  are  not 
common. 

RULE. 

Multiply  the  denominators  together  for  the  new 
denominator. 

Subtract  the  denominators,  and  write  the  differ- 
ence over  the  new  denominator.  Reduce^  if  possible. 
Hie  fraction  thus  formed  will  be  the  answer  required. 

EXAMPLES. 

1.  From  J  take  -J.  Ans.  J  . 

OPERATION. 

Multiply  the  denominators  8  and  4  :  8  times  4 
are  32,  the  new  denominator.  Subtract  the  denomi- 
nators :  4  from  8  leaves  4,  the  new  numerator.  The 
fraction  is  now  -^-,  which  reduced,  by  dividing  both 
numerator  and  denominator  by  4,  4)-g^=£,  the  re- 
quired answer. 

2.  From  -J-  take  |.  Ans.  -fs. 

3.  From  I  take  T1¥.  Ans.  -fc. 


196  INTERMEDIATE   ARITHMETIC. 

4.  From  ^  take  T1T.  Ans. 

5.  From  TV  take  TV.  Ans.  T^. 

6.  From  -fa  take  -Jj.  .4ns.  -j-J-j-. 

7.  From  TV  take  ^  ^4rcs.  y^. 

8.  From  y1^  take  TV.  ^ws.  -g-f  0. 

9.  From  ytg-  take  TV  ^»s.  ^¥. 

10.  From  Jg-  take  T1T.  Ans. 

11.  From  J    take   L.  Ans. 


LESSON  XVI. 

CASE  III. 

.  When  the  numerators  are   other  figures 
than  1. 

RULE. 

Find  the  least  common  multiple  of  the  denom- 
inators. 

Find  the  numerators  by  dividing  the  common 
multiple  by  the  denominator  and  multiplying  the  nu- 
merators by  the  quotient.  Subtract  the  numerators, 
and  write  their  difference  over  the  common  multiple. 
Reduce,  if  possible.  ^ 

EXAMPLES. 

1.  From  £  take  |. 

OPERATION. 

20-15         5 


24  24 


The   common   multiple   between  6  and  8  is  24, 
the  new  denominator.     Divide  24  by  C,  one  of  the 


SUBTRACTION   OF   FRACTIONS.  197 

numerators ;  the  quotient  is  4.  Multiply  the  numer- 
ator 5  by  4  ;  the  product  is  20.  Write  20  above  the 
line.  Divide  the  multiple  24  by  8 ;  the  quotient  is 
3.  Multiplying  the  numerator  5  by  3  gives  15. 
Write  15  as  above.  Subtract  the  numerator  15  from 
the  numerator  20,  and  write  the  difference  over  the 
common  multiple. 

2.  From  f  take  J.  Am.  -f. 

3.  From  -J  take  -fa.  ^.ns.  f . 

4.  From  T%-  take  -J -£.  *Am.  J. 

5.  From  f«-  take  ^_.  ^ws<  « ]. 

6.  From  {4  take  T%. 

7.  From  |f  take  £. 

8.  From  ||-  take  f . 

9.  From  J-f  take  ^8. 

10.  From  ||  take  TV  Am.  $ft. 

11.  From  |f  take  -&.  ^4?zs.  J{>-J. 

12.  From  f|  take  T%.  Am.  f*-f. 

13.  From  f|  take  ^.  Jw«.  .1  .jj ;. 

14.  From  \ %  take  T%.  Am.  fj. 

15.  From  |f  take  T%.  ^W5.  -J  J, 

16.  From  ££  take  T\.  ^W5.  T5T. 

17.  From  |-|  take  JJ.  ^ws.  |j|. 

18.  From  |J  take  |.  ^w*.  J. 

19.  From  ||-  take  T\.  ^«5.  J». 

20.  From  f£  take  ^.  ^«5.  JJ. 

21.  From  |-|  take  f.  Am.  f?, 

17* 


198  INTERMEDIATE   ARITHMETIC. 


LESSON  XVII. 

CASE  IV. 

When  one  mixed  number  is  to  be  subtract- 
ed from  another. 

EULE. 

Reduce  the  mixed  numbers  to  improper  fractions. 
Find  the  least  common  multiple  of  the  denominators. 
Divide  the  least  common  multiple  by  the  denominator, 
and  multiply  the  numerator  of  the  improper  fraction 
by  the  quotient.  Subtract  the  numerators,  and  write 
the  difference  over  the  common  multiple.  Reduce,  if 


EXAMPLES. 

1.  From  12 £  take  5|. 

OPEEATIOX. 

Reducing  the  mixed  numbers  to  improper  frac- 
tions, gives  J-jp-  and  ££.  The  least  common  multiple 
between  8  and  6  is  24,  the  new  denominator. 

303  —  116 


24 

8  into  24,  3  times ;  3  times  101  gives  303,  the 
new  numerator.  6  into  24,  4  times ;  4  times  29  gives 
116,  the  other  new  numerator.  Subtracting  116 
from  303  leaves  187.  Writing  187  over  the  least 
common  multiple,  24,  gives  -V8^  for  the  required 
fraction.  Reducing  -y^7-  to  a  mixed  number  gives 
7|-f,  the  required  answer. 


SUBTRACTION   OF   FRACTIONS.  199 

2.  From  12 1  take  4£.  Ans.  s  ,V 

3.  From  13£  take  5|.  -4wa.  8^j. 

4.  From  14i  take  6-J.  ^/z*.  7?*. 

5.  From  15-jj-  take  7^.  -4w*.  7|£. 

6.  From  16f  take  8f!j,  ^ws.  8,:Yv 

7.  From  17 -g-  take  9f.  Ans.  S-f{. 

8.  From  IS*  take  lOf. 

9.  From  19TVtake  11-^-. 

10.  From  20|a  take  12»,  Ans.  8^ 

11.  From  21f^  take  13-^.  ^iws.  8^5- 

12.  From  22f|  take  14^.  Ans.  8TJT 

13.  From  23||  take  15|^.  -4«*.  8-b^ 

14.  From  24f|  take  16f|.  ^fws.  8TfT 

15.  From  25|f  take  17-J-f.  ^4w*. 

16.  From  26f«  take  18{*.  -4ws. 

17.  From  27|-£  take  19||.  Ans. 

18.  From  28|f  take  20-J~|.  -4w*.  83 

19.  From  29f|  take  21f|. 

20.  From  30ff-  take  22||.  Ans.  S-*iff 

21.  From  32f£  take  23 Jf. 


LESSON  XVIII. 

CASE   V. 

When  a  mixed  number  is  to  be  subtracted 
from  a  whole  number. 

RULE. 

Write  the  whole  number  above  the  mixed  number, 
as  in  simple  subtraction,  with  the  fractional  part  a 
little  to  the  right  and  not  under  the  whole  number. 


200 


INTERMEDIATE   ARITHMETIC. 


Borrow  1  from  the  whole  number,  ichich  is  equal 
to  as  many  of  the  same  denomination  as  the  fraction 
below  as  would  make  a  whole  one. 

Subtract  the  fractional  parts,  and  write  the  differ- 
ence over  the  common  denominator. 

Add  1  to  the  first  figure  of  the  subtrahend  and  sub- 
tract the  figures  as  in  simple  subtraction. 


EXAMPLES. 

1.  From  12  subtract  2J. 

Borrow  1  from  12,  which  is 
equal  to  |-.  Subtracting  j|  from  | 
leaves  |  for  the  fractional  part. 
Add  1  to  2,  making  3.  3  from  12 
leaves  9.  The  answer  required  is  9-jj-. 

Or: 


OPERATION. 
12 


9|-    Ans. 


o  Subtract  the  numerator  of  the  fraction 
from  the  denominator,  and  write  the  difference  over 
the  denominator. 

Add  1  to  the  first  figure  of  the  subtrahend,  and 
subtract  as  in  simple  numbers. 


EXAMPLE. 

Subtract  the  numerator  3  from 
the  denominator  8,  and  the  remain- 
der will  be  5.  Write  the  5  above  the 
8,  as  in  the  example.  Add  1  to  the 
figure  2,  making  it  3.  3  from  12 
leaves  9.  The  answer  is  9j>. 


OPERATION. 
12 


SUBTRACTION   OF   FRACTIONS.  201 

Or: 

238.  Consider  the  denominator  of  the  whole 
number  to  be  1  ;  and  the  sum  is,  from  ±f-  subtract 
2-J.  Reducing  2|  to  an  improper  fraction  gives  ^ 
The  sum  now  stands,  From  •*-*  take  -y. 

96  —  19 


8 

The  common  multiple  is  8.  1  into  8,  8  times.  Mul- 
tiplying the  numerator  12  by  8  gives  96  for  the  new 
numerator.  8  into  8,  once;  once  19  is  19,  the  other 
numerator.  Subtracting  19  from  96  leaves  77,  which 
written  over  the  numerator  8  gives  -^T.  Reducing 
-^  to  a  mixed  number  gives  (8  into  77,  9  times  and 
5  remaining),  9|,  the  required  answer. 

2.  From  18  take  12-f.  Ans.  5-J-. 

3.  From  19  take  13f  Ans.  5-*-. 

4.  From  20  take  14f.  Ans.  5-J-. 

5.  From  21  take  15^.  Ans.  5^. 

6.  From  22  take  16f£.  Ans. 

7.  From  23  take  17|-|.  Ans. 

8.  From  28  take  12f.  Ans.  15f 

9.  From  30  take  13^.  Ans.  16ff. 

10.  From  36  take  14^f.  Ans.  21T\. 

11.  From  42  take  5fi.  Ans.  36^. 

12.  From  89  take  16-f  J.  Ans.  72 JJ. 

13.  From  34  take  17jf.  Ans.  16^. 

14.  From  98  take  18|f  Ans.  79||. 

15.  From  240  take  19^|.  Ans.  220£f. 

16.  From  380  take  20f|.  Ans.  359^. 


202 


INTERMEDIATE  ARITHMETIC. 


17.  From  760  take  21|-|.  Ans.  738^- 

18.  From  123  take  22-fjj-.  Ans.  lOOf-f. 

19.  From  345  take  lOOff.  Ans.  244f|. 

20.  From  6789  take  2000ff|.  Ans.  47S8|. 

21.  From  101112  take  1897f°-f  Ans-  99214f|. 


LESSON  XIX. 

CASE   TI. 

239.  When  a  whole  number  is  to  be  subtracted 
from  a  mixed  number. 

RULE. 

Write  the  fraction  below  the  line  to  the  right  oj 
the  remainder.  TJie  remainder  is  found  by  subtract- 
ing as  in  simple  numbers. 


EXAMPLES. 

1.  From  141  take  5. 

Take  the  fractional  part  of  the 
mixed  number,  and  write  it  below 
the  line  as  in  the  example. 

Subtract  5  from  14,  leaving  9,  the 
answer  is  9|. 

2.  From  14f  take  12. 

3.  From  13-jL  take  2. 

4.  From  18f  take  8. 

5.  From  22f  take  10. 

6.  From  46|  take  21. 

7.  From  98^  take  64. 


OPERATION. 


9|  Ans. 


Ans.  2-f. 
Ans.  11£. 
Ans.  lOf. 
Am.  12|. 
Ans.  25f . 
Ans. 


SUBTRACTION    OF   FRACTIONS.  203 

8.  From  100f£  take  25.  Ans.  75f$. 

9.  From  286JJ  take  89.  Ans.  197}. 

10.  From  300ff  take  98.  Ans.  202=  j. 

11,  From  496T2o2T  take  248.  Ans.  248-122f. 

LESSON  XX. 

CASE  VII. 

240.  When  one  mixed  number  is  to  be  subtract- 
ed from  another,  the  fractional  part  of  the  subtrahend 
being  greater  than  the  fractional  part  of  the  min- 
uend. 

EULE    I. 

Reduce  the  mixed  numbers  to  improper  fractions, 
and  subtract  as  in  subtraction  of  fractions. 

Or: 

EULE   II. 

241 .  To  the  fractional  part  of  the  minuend  add 
1  whole  one.    Reduce  the  1  whole  one  and  the  frac- 
tion to  a  mixed  number.     You  then  have  a  pro- 
per fraction  to  be  subtracted  from  an  improper  one. 

Subtract  the  fractional  parts  as  in  subtraction  of 
fractions. 

Add  1  to  the  1st  figure  of  the  dividend,  and  sub- 
tract as  in  simple  numbers. 

EXAMPLES. 

1.  From  12 J  take  10$. 


204  INTERMEDIATE   ARITHMETIC. 

Reducing  to  improper  fractions  gives 

37      d  21  _  74  —  63  _  11  _     g 
3  2  6  6 

The  least  common  multiple  is  6. 

The  numerators  are  74  and  63. 

Subtracting  63  from  74  leaves  11,  which  written 
over  the  multiple  6,  gives  *£-  for  the  required  frac- 
tion. Reducing  ±£-  to  a  mixed,  gives  If,  the  re- 
quired answer. 

Or: 

OPERATION.  l.  From  12-1-  take  10-|-. 

£  being  larger  than  -J-,  it  cannot 
be  subtracted.     Borrow  1  whole  one 
from  the  12,  making  f,  together  with 
%   already   in    the    minuend,    would 
make  $.     The  sum  now  would  read,  from  11$  take 


|-  is  equal  to  -f ,  and  %  is  equal  to        OPEEATIOX. 
f.     Write  the  8  a  little  to  the  right  4  _ 

of  the  $,  and  write  the  3  a  little  to 
right  of  the  -|. 

The  denominator  6  is  understood 
to  be  written  under  the  8  and  3.     Subtracting  3  from 
8  leaves  5,  that  is,  •§-.     Write  the  -f-  under  the  frac- 
tional parts,  and  subtract  the  10  from  the  11.     10 
from  11  leaves  1. 

The  required  answer  is  1-f-. 


SUBTRACTION   OF   FRACTIONS.  205 

2.  From  2J  take  If.  Am.  -§ . 

3.  From  2 1  take  If  Am.  £. 

4.  From  3£  take  1-J.  ^HS.  1J. 

5.  From  31  tade  If.  Am.  1  £. 

6.  From  4-J-  take  2T3F.  ^«s.  1  ,"„. 

7.  From  4|  take  3^-.  Am.  -*. 

8.  From  4|  take  1-J-f.  -4w*.  2-J. 

9.  From  5£  take  2£.  Am.  2£. 

10.  From  5|  take  3f .  Am.  \\ ;"]. 

11.  From  5|  take  4^.  Am.  TV 

12.  From  GJ  take  5-f.  ^^5.  4. 

13.  From  Of  take  If.  u4w*.  4f 

14.  From  Gf  take  If.  Am.  4%*. 

15.  From  V-J-  take  3|.  ^(ws.  3|. 

16.  From  7|  take  4-J.  Am.  2£. 

17.  From  7|  take  3|f  ^«5.  3{|. 

18.  From  8^  take  5^.  Am.  2fJ. 

19.  From  8|  take  6T6g.  Am.  If. 

20.  From  8|  take  7^.  ^ws.  |J. 

21.  From  9^  take  4^.  Am.  4ijJ. 

22.  From  9T3¥  take  5^.  Am.  3|-»-. 

23.  From  9TV  take  6-|f.  ^«s.  2f|. 

24.  From  12  j  take  7|.  ^ws.  4|, 

25.  From  13J-  take  8|.  Am.  4f. 

26.  From  21TV  take  9T3^.  Am.  llf. 

27.  From  31-^  take  12T2T.  Am.  18f}. 

28.  From  14^-  take  11T2¥.  Am.  2-ff. 

29.  From  15^-  take  12¥V  Am.  2fg. 

30.  From  41f-J  take  21ff.  Am.  19f», 

31.  From  51|~|  take  3 Off.  Am. 

18 


206 


INTERMEDIATE   ARITHMETIC. 


MULTIPLICATION  OF  FRICTIONS. 

LESSOR  XXI. 

PEINCIPLES    01?   CANCELLING. 

242.  From  what  we  have  already  seen,  we  know 
that  dividing  the  numerator  and  denominator  by  the 
same  number  does  not  alter  the  value  of  the  frac- 
tion.    The  form  alone  is  changed,  not  the  value. 

243.  In    multiplication    of   fractions   we    may 
consider  the  several  fractions  of  which  a  sum  is  com- 
posed as  one  compound  fraction,  because  the  word 
"of"  means  multiplication.     Now,  if  we  divide  the 
numerator  of  any  one  of  those  fractions  by  any  num- 
ber, and  divide  the  denominator  of  any  of  the  other 
fractions  by  the  same  number,  the  value  of  the  mul- 
tiplication is  not  changed.     Thus, 


If  we  divide  the  numerator  (6)  of  the  second 
fraction  by  3,  and  divide  the  denominator  (3)  of  the 
first  fraction  by  3,  the  value  of  the  multiplication  is 
not  changed.  We  get  2  for  the  numerator  of  the 
second  fraction,  and  1  for  the  denominator  of  the  1st 
fraction.  These  remarks  apply  to  any  number  of 
fractions. 


MULTIPLICATION    OF  FRACTIONS.  207 


n 

U 


Divide  (3)  denominator  and  (6)  numerator  by 
3.  Divide  (2)  numerator  and  (8)  denominator  by  2. 
Dixide  (14)  numerator  and  (4)  denominator  (quo- 
tient) by  2.  Divide  (7)  denominator  and  (7)  numer- 
ator (quotient)  by  7.  Divide  (2)  numerator  (quo- 
tient) and  (2)  denominator  (quotient)  by  2.  There 
remain  but  1's  for  the  numerator,  and  1's  for  the 
denominator.  That  is,  all  the  factors  of  the  numer- 
ators are  common  and  equal  to  those  of  the  denom- 
inators. 

PKOOF. 
The  factors  of  the  numerators  are  as  follows  : 

2  =  2  14  =  2X7 
6  =  2X3 

The  factors  of  the  denominators  are  : 

3  =  3  8  =  2X2X2 

V  =  7 

"Writing  the  factors  of  the  numerators  and  de- 
nominators as  one  fraction,  with  the  sign  of  multi- 
plication between  them,  we  have 

2x2x3x2xY 
3x7x2x2x2 


208  INTERMEDIATE  ARITHMETIC. 

Cancelling  the  factors  common  to  both,  gives 
11111 


$x#x#x#x2 
11111 

1's  for  the  numerator,  and  1's  for  the  denominator. 

244.   MULTIPLICATION    OP   FRACTIONS. 
CASE  I. 

GENERAL    RULE. 
Multiply  all  the  numerators  together  for  a  new 
numerator,  and  all  the  denominators  together  for  a 
new  denominator,  after  having  cancelled  all  the  fac- 
tors common  to  both. 

245.  When  a  fraction  is  to  be  multiplied  by  a 
whole  number : 

RULE. 

Multiply  the  whole  mimber  by  the  numerator,  and 
divide  the  product  by  the  denominator.  If  any  re- 
mainder, write  it  in  the  form  of  a  fraction. 


EXAMPLE. 


Multiply  f  by  8. 


OPERATION. 


2    X 


Multiply  the  8  by  the  de- 
nominator 2,  and  write  the 
result  over  the  denominator 
in  the  form  of  an  improper 
fraction,  and  reduce  the  improper  fraction  to  a  mixed 
number. 


MULTIPLICATION   OF   FRACTIONS.  209 

EXEECI3ES. 

1.  Multiply  J  x  4.  Ans.  2. 

2.  Multiply  f  x  6.  Ans.  4. 

3.  Multiply  f  x  8.  Ans.  6. 

4.  Multiply  f  x  10.  Ans.  8. 

5.  Multiply  f  x  12.  Ans.  10. 

6.  Multiply  f  x  14.  Ans.  12. 

7.  Multiply  |  x  23.  u4«s.  20£. 

8.  Multiyly  f  x  98.  Ans.  87£. 

9.  Multiply  T°o-xl8.  Ans.  ICt. 

10.  Multiply  |f  x  24.  Ans.  21 -ft. 

11.  Multiply  •}•£  x  36.  -4ftS.  33. 

12.  Multiply  |f  x  39.  Ans.  3G. 

13.  Multiply  f|x98.  Ans.  91. 

14.  Multiply  ££x72.  Ans.  6Vf 

15.  Multiply  |f  x  100.  yltts.  93|. 

16.  Multiply  ff  x  156.  Ans.  146ff. 

LESSON  XXII. 

346.  When  a  mixed  number  is  to  be  multiplied 
by  a  whole  number : 

KULE. 

Reduce  the  mixed  number  to  an  improper  fraction. 
Multiply  the  numerator  thus  found  by  the  whole  num- 
ber and  divide  the  result  by  the  denominator. 

EXAMPLE. 

Multiply  2  J  by  8. 

3          8         V  x  8       56 

y    T:  ~Y-  ZT~- 

18* 


210 


INTERMEDIATE   ARITHMETIC. 


Reducing  2^  to  an  improper  fraction  gives  J. 
The  problem  is,  as  it  now  stands  :  Multiply  J  by  8. 
Multiply  the  numerator,  7,  by  the  whole  number,  8, 
and  divide  the  product  (56)  by  the  denominator,  3. 
If  any  remainder,  write  it  in  the  form  of  a  fraction. 


EXERCISES. 

1.  Multiply  21  by  18. 

2.  Multiply  3-J-  by  20. 

3.  Multiply  4£  by  22. 

4.  Multiply  6J  by  24. 

5.  Multiply  9|  by  40. 

6.  Multiply  10fJ  by  30. 

7.  Multiply  Hfi-by  20. 

8.  Multiply  21if  by  12. 

9.  Multiply  62f  by  8. 

10.  Multiply  18fby  10. 

11.  Multiply  31$  by  11. 

12.  Multiply  46fby  12. 

13.  Multiply  92f  by  13. 

14.  Multiply  100J  by  14,  15,  17,  18,  19,  12,  21. 

15.  Multiply  131-iW  by  22,  23,  24,  25,  26,  27,  28. 


Ans.  40£, 
Ans.  64. 
Ans.  1061 
Ans.  165. 
Ans. 
Ans. 

Ans.  2331. 
Ans.  263T7g-. 
Ans.  501J. 
Ans.  187f 
Ans.  349$. 
Ans.  562. 


LESSOR  XXIII. 

S47.  To  multiply  a  whole  number  by  a  fraction. 
RULE. 

Multiply  the  whole  number  by  the  numerator  of 
the  fraction  and  divide  the  product  by  the  denomina- 
tor of  the  fraction. 


MULTIPLICATION   OF   FRACTIONS.  211 

EXAMPLE. 

Multiply  21  by  £ 

OPERATION. 


EXERCISES. 

1.  Multiply  10  by  £.  Ans.  5. 

2.  Multiply  20  by  |.  ^4rcs.  13£. 

3.  Multiply  24  by  f.  Ans.  18. 

4.  Multiply  28  by  -j,  ^t??s.  22~. 

5.  Multiply  32  by  |.  Ans.  26|. 

6.  Multiply  36  by  f.  .4«s.  30f  . 

7.  Multiply  40  by  -J.  ^1?2S.  35. 

8.  Multiply  44  by  |.  ^4n5.  39  J. 

9.  Multiply  48  by  T%. 

10.  Multiply  52  by  ^, 

11.  Multiply  56  by  ||.  ^715.  51-J-. 

12.  Multiply  60  by  |f.  ^l/zs.  55^. 

13.  Multiply  64  by  |f.  Ans.  59|. 

14.  Multiply  68  by  ||.  ^4^5.  63T\. 

15.  Multiply  72  by  J-f  .  ^t^s.  30. 

16.  Multiply  74  by  JJ.  -4ws.  37. 

17.  Multiply  78  by  ||.  ^^5.  36. 

18.  Multiply  82  by  \.  Ans.  41. 

19.  Multiply  86  by  f  .  Ans.  57£. 

20.  Multiply  100  by  -fa.  Ans.  8. 


212  INTERMEDIATE    ARITHMETIC. 


LESSON  XXIV. 

248.  To  multiply  one  simple  fraction  by  an- 
other. 

RULE. 

Multiply  all  the  numerators  together  for  a  new 
numerator,  and  all  the  denominators  together  for  a 
new  denominator,  after  having  cancelled  all  the  fac- 
tors common  to  both. 

EXAMPLE. 
T%T    TJ.'    ^       i        o        «        JL  X  JS  X  O          O        . 

Multiply  i  x  f  x  4—s — o — *  —  ^r  -d-ns- 

By  cancelling  the  figure  2  in  the  numerator  an< 
denominator,  there  remains  but  1  and  5  in  the  nu- 
merator, and  3  and  7  in  the  denominator.  Multiply- 
ing the  1  by  the  5,  gives  5  for  the  numerator  of  the 
answer,  and  by  multiplying  the  3  and  the  7,  we  get 
21  for  the  denominator.  The  answer  is,  then,  /T. 

EXEECISES. 

1.  Multiply  i  by  f  off.  Ans. 

2.  Multiply  I  by  T%  oi'*£-.  Ans.  f. 

3.  Multiply  |  by  ff  of  ff  Ans.  |. 

4.  Multiply  -|  by  f  off.  Ans.  -|. 

5.  Multiply  |  by  f  off.  Ans.  -J. 

6.  Multiply  iV  by  T9g 

7.  Multiply  f!>-  by  f|  of 


MULTIPLICATION    OF    FRACTIONS.  *J  1  •> 


8.  Multiply  H  by  f§  oH  £. 

9.  Multiply  If  by  f«-  of  -=}  jj, 

10.  Multiply  |J  by  ff  of  ff  .  Ana.  \\. 

11.  Multiply  Jf  by  ft  of  |f  ^»«.  T\. 

12.  Multiply  |f  by  |£  of  -Jg-.  -4w«.  ff 

13.  Multiply  |§Xi-JX|2-S-Xffr  ^w«-  L 

14.  Multiply  f|Xf|XVXf  ^»*.  1. 

15.  Multiply  iixS{x55«xW-  ^ws-  !• 

16.  Multiply  HXiVrXtXA.  ^W5-  A- 

17.  Multiply  f{|XT8e0oXiXf 

18.  Multiply  |  §X-JfX-j-Xj-. 

19.  Multiply  fjxllfXjtXt.  -4»«.  f 

20.  Multiply 

21.  Multiply 

22.  Multiply 

23.  Multiply 

24.  Multiply 

25.  Multiply 

26.  Multiply 
27. 

28.  Multiply  -UXWXVXA-  An8'  4- 

29.  Multiply  T^XW-XV-XyV  -4««.  4. 

30.  Multiply 


LESSON  XXV. 

.  To  multiply  one  mixed  number  by  an- 
other. 

E  U  L  E  . 

Reduce  the  mixed  numbers  to  improper  fractions, 
and  multiply  the  numerators  together  for  a  new  nu~ 


214 


INTERMEDIATE   ARITHMETIC. 


numerator,  and  multiply  the  denominators  together 
for  a  new  denominator,  offer  having  cancelled  the 
factors  common  to  both  numerator  and  denominator. 


EXAMPLE. 


Multiply  2£  by  34=5  x—  =  -f 

J2       3         3 

1st.  Reduce  the  mixed  numbers  to  improper  frac- 
tions. 2d.  Cancel  the  denominator  2,  and  the  nu- 
merator 10,  by  dividing  both  by  2.  3d.  Multiply  the 
numerator  5,  and  the  quotient  5  together,  and  write 
the  product  over  the  denominator  3,  in  the  form  of 
an  improper  fraction.  Reduce  the  improper  fraction 
to  a  mixed  number,  by  dividing  the  numerator  by 
the  denominator,  and  if  there  is  a  remainder,  write  it 
in  the  form  of  a  fraction. 


EXEECISES. 


1.  Multiply  1J-  by  2|. 

2.  Multiply  2|  by  3}. 

3.  Multiply  3 1  by  4-f. 

4.  Multiply  4|  by  5f. 

5.  Multiply  5-f  by  6-f. 

6.  Multiply  6f  by  7-J. 
*7.  Multiply  7-J-  by  8-f . 

8.  Multiply  8|  by  9T9T 

9.  Multiply  9T9¥ 

10.  Multiply 

11.  Multiply 


Am.  4. 
10. 
Ans.  18. 
.  28. 
.  40. 
Ans.  54. 
-4w*.  70. 
Ans.  88. 
.  108. 
.  130. 
Ans.  154. 


DIVISION   OF   FRACTIONS.  U  1  ."> 


12.  Multiply  12ff  by  13{;-.  ^4^.  ISO. 

13.  Multiply  13-J-J  by  14  {  1.  Ans.  208. 

14.  Multiply  14{f  by  15]jj-.  ^Ln«.  238. 

15.  Multiply  15j|by  IG'-f.  -4w&  270. 
1C.  Multiply  1G-14  by  17l,v  Ans-  304- 

17.  Multiply  17f|by  18f§.  Ans.  340. 

18.  Multiply  20  J-  by  10J-.  Ans.  21  !-«-. 

19.  Multiply  30  1  by  15J.  ^l?w.  46  7|. 

20.  Multiply  40}  by  20-}-.  ^4/w.  821ff. 

21.  Multiply  50-jj-  by  25f.  Ans.  1290^. 

22.  Multiply  GO*  by  30  J.  Ans.  18371. 

23.  Multiply  70«  by  35,s().  ^^,9.  251Gj>. 

24.  Multiply  SO^by  40Tfir.  Ans.  3255-f. 

25.  Multiply  90^-  by  45-^.  Ans.  4130||. 

26.  Multiply  100^/v  by  50TV  Ans.  5075J. 

27.  Multiply  HOj^by  55^.  Ans.  6132}. 

28.  Multiply  112/g-  by  56  J.  ^4/w.  G356JL. 
2  9.  Multiply  1  1  3T»8-  by  1  :  :  1  1]  .  ^>is.  1  5  :  !  •_'  !  . 
30.  Multiply  114fg-  by  57^  Ans.  C583}. 


DIVISION  OF  FEACTMS. 
LESSON  XXVI. 

CASE   I. 

.  To  divide  a  whole  number  by  a  fraction. 

RULE. 
Invert  the  terms  of  thz  divisor  and  multiply. 


216 


INTERMEDIATE   AEITHMETIC. 


Divide  4  by 


EXAMPLE. 


OPEEATIOX. 


-  =  6,     Ans. 


EXERCISES. 


1.  Divide 

2.  Divide 

3.  Divide 

4.  Divide 

5.  Divide 

6.  Divide 

7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 

12.  Divide 

13.  Divide 

14.  Divide 

15.  Divide 

16.  Divide 

17.  Divide 

18.  Divide 

19.  Divide 

20.  Divide 

21.  Divide 

22.  Divide 

23.  Divide 

24.  Divide 


10  by  f. 
12  by  f 

14  by -ft. 

16  by  f . 
18byTV 
20  by  -ff. 
22  by  ff. 
24  by  if. 
26  by  -ff. 
36  by  U. 
46  by  |f. 
56  by  |f. 
66  by  ft. 

^  *y  T9o6o 

86  by  f  o, 

96  by  ,V 
106  by  T 
206  by  T 
306  by  f . 
406  by  |. 


Ans.  6. 
20. 

42. 

Ans.  64. 
^Ins.  15. 

Ans.  27. 
.  18f. 
22  f. 
Ans.  33. 


Ans.  234. 
.  2S|. 
.  27-f. 
Ans.  37f. 
^^5.  48^. 
Ans.  58i|. 
.  36. 
.  79-J, 
Ans.  172. 
^Z6'.  1728. 
Ans.  381-f. 
.  412. 
.  459. 
45  6  J. 


DIVISION    OF    FRACTION'S.  217 


LESSON  XXVII. 

CASE   II. 

251.  To  divide  a  fraction  "by  a  whole  number. 

RULE. 

Consider  the  whole  number  to  be  the  denominator 
of  a  fraction  whose  numerator  is  1.  Then  multiply 
as  in  multiplication  of  fractions. 

EXAMPLE. 

Divide  f  by  4. 

OPERATION. 

T^4=4xi  =  !'  Ans- 

2 

EXERCISES. 

1.  Divide  f  by  12.  Ans.  -fa. 

2.  Divide  f  by  3.  Ans.  £. 

3.  Divide  $  by  14.  Ans.  ^. 

4.  Divide  -f  by  5.  Ans.  -J-. 

5.  Divide  -f  by  16.  Ans.  -f^. 

6.  Divide  -J  by  7.  ^bis.  -*. 

7.  Divide  f-  by  18.  Ans.  -ti\. 

8.  Divide  T9y  by  9. 

9.  Divide  ^  by  20. 

10.  Divide  fj-  by  11.  Ans.  TV 

11.  Divide  f|  by  22.  Ans.  T-|T. 

12.  Divide  |£  by  13.  Ans.  ^. 

13.  Divide  |f  by  24.  Ans.  yj^. 

19 


218  INTERMEDIATE   ARITHMETIC. 

14.  Divide  J-f-  by  15.  Ans.  -Jg-. 

15.  Divide  |f  by  26.  Ans.  -g|T. 

16.  Divide  fj  by  17.  ^tws.  flg. 

17.  Divide  |f  by  28.  ^ws.  ^jj^. 

18.  Divide  f§-  by  19.  Ans.  fa. 

19.  Divide  ff  by  40. 

20.  Divide  f|  by  22. 

21.  Divide  ff  by  42.  ^.ws. 

LESSON  XXVHL 

CASE  III. 

252.  To  divide  one  simple  fraction  by  another. 

EULE. 

Invert  the  terms  of  the  divisor,  and  multiply  as 
in  multiplication  of  fractions. 

EXAMPLE. 

1.  Divide  fbyf  -f:!  —  fxf  —  f-f  Ans. 

Invert  the  terms  of  the  divisor  |- ,  and  multiply. 


2.  Divide  $  by  ff  ^Iws.  f  J-. 

3.  Divide  f  by  i|.  Ans.  ff 

4.  Divide  |  by  f|.  ^lw,s.  if. 

5.  Divide  -|  by  f|. 


6.  Divide  f  by  ff.  . 


7.  Divide  f  by  f|.  ^Iws.  ff  . 

8.  Divide  |  by  ff.  Ana.  \^. 

9.  Divide  f  by  f|- 


DIVISION    OF   FEACTIONS.  219 

10.  Divide  -^  by  ff.  Ans.  f|. 

11.  Divide  {f  by  ff.  Ans.  {%%. 

12.  Divide  \\  by  [;{.  ^4?*s.  ^f. 

13.  Divide  }|  by  f|.  Ans.  ly^-j. 

14.  Divide  J--J  by  \\.  Ans.  Ijf-^. 

15.  Divide  \\  by  /„. 

16.  Divide  J{  by  $. 

17.  Divide  tf  by  f. 

18.  Divide  Jj  by  f . 

19.  Divide  f§  by  ||.  ^l^s.  lf|. 

20.  Divide  J-g-  by  ^.  ^4?i5.  1^.-. 

21.  Divide  ff  by  f.  Ans.  IfJ. 

LESSON  XXIX. 

CASE  IV. 

353.  To  divide  one  mixed  number  by  another. 

RULE. 

Reduce  the  mixed  numbers  to  improper  fractions. 
Invert  the  terms  of  the  divisor  and  multiply. 

EXAMPLES. 

1.  Divide  2$  by  3J. 

1  ol  _5      .      10   _   #  3      _3 

2  •  °-s—  g   •     3   —  2  X  10  ~"  4 

2 

Reducing  2^  and  3^  to  improper  fractions  gives 
|  and  ^3°-.  Inverting  ^3°-,  cancelling  and  multiplying, 
gives  f . 


220 


INTERMEDIATE   ARITHMETIC. 


2.  Divide 

3.  Divide 

4.  Divide 

5.  Divide 

6.  Divide 

7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 

12.  Divide 

13.  Divide 

14.  Divide 

15.  Divide 

16.  Divide 

17.  Divide 

18.  Divide 

19.  Divide 

20.  Divide 


2-s  by  20£. 
31  by  191. 
4|  by  18}. 
5fbyl7f 
61  by  161 
7|  by  15|. 
8£  by  14J. 
9A  by  13TV 
10A  by  12A- 

11 A  by  nT5if- 
12A  ^  10  A- 
ISA  fcy  »&• 

14fV  by  8^. 

ISA  by  VA- 

16A  by  6T3¥. 

1?  A  by  5A- 
18«by4».- 
19Mby3|^. 
20|  by  2|. 


Ans. 
Ans. 


Ans. 


Ans. 


COMPLEX  FRACTIOHS. 

LESSON  XXX. 

354.  A  complex  fraction  is  one  which  has  a 
fraction  for  its  numerator,  or  a  fraction  for  its  de- 
nominator, or  a  fraction  for  both,  Thus,  -^-,  f ,  ~ 

¥     4     f 
are  complex  fractions. 


COMPLEX  FEACTIONS.  221 

ADDITION   OF   COMPLEX   FRACTIONS. 
EXAMPLE. 


...  5        2 

Add  f  +T=lTxirxx^ 

2 


The  first  complex  fraction  is  read  f  divided 
by  -f,  the  second  \  divided  by  f  ;  from  which 
we  have  two  problems  in  division.  By  perform- 
ing the  first  division  according  to  the  rule  of 

rt  K  K 

division,  we  have  —  x  —r  or  -7-.     Applying  the  rule 
3        fa.          0 

2 
of   division    to   the  second   complex   fraction,   we 

2 

1       &         2 

have  -r-  x  —  or  —  .     The   question   is  now  reduced 
ft        3          3 

to  f  -f  f.     According  to  the  rule  of  addition  of  frac- 
tions, we  find  --  or  —  ,  which,  reduced  to  a  mixed 

number,  gives  1—,  or  li  for  the  answer. 


i.  Add 


iA  and  (JL)  AnSt  11 


2.  Add  and  (±).  An,. 


3.  Add          and         . 


222 


INTERMEDIATE   ARITHMETIC. 


4.  Add    4-    and  M? 


7.  Add 

wv 


ffl-ffli 


LESSON  XXXI. 

SUBTRACTION   OF  COMPLEX  FRACTIONS. 
EXAMPLE. 


From       take      . 

2  d 

First  reduce  the  mixed  numbers  to  improper 

1JL  5. 

fractions.     From  ~  take  -|-.     Place  the  complex 

¥  ¥ 

fractions  in  the  form  of  division  of  fractions  : 
(JJL_J_I)_  (A-:-!).  Inverting  the  terms  according  to 
the  rule  of  division  of  fractions,  and  cancelling: 

(—  x  —  )  —  f—  X  -ir).  Multiply  as  in  multiplica- 
\'3  I/  \3  tp  / 

O  O  Q 

tion  :  -  --  —  -.     Perform  the  subtraction  according 
3          3 


to  the  rule  of  subtraction  : 


28-8 


20 
~3 


=  —  =  6}.     Ans. 


G.  From -|  take    gf 


COMPLEX  FKACTIONS.  223 

1.  From  —  take  —-.  Am.  9  J  J. 

2  3" 


2.  From  — |  take  •—.  Am. 

184  9* 

3.  From  -  --  take  -f . 

2  6" 


64|  14f 

V.  From  — ^  take -— -7.  ^f«5. 


8.  From  -    ^  take  ^f . 

54  8t 


LESSON  XXXII. 

MULTIPLICATION   OF   COMPLEX   FKACTIOXS. 
EXAMPLE. 

Multiply  |  by  |. 

4  A_» 

Reduced  to  improper  fractions  gives  -{ -  by  -j. 

T  3^ 

Inverting  the  denominators  gives  |  x  f  by  -^  x  T33. 


224 


INTERMEDIATE  ARITHMETIC. 


Since  the  word  by,  in  this  sum  means  multiplication, 
substitute  the  sign  of  multiplication  in  its  place,  and 
the  fraction  will  be  a  continued  multiplication  ;  thus, 


49 


^T    X    —    X    —    X 
$  9  b 


— .     Multiplying   the   numerators 

and  denominators  gives  -J-j^f ;  which,  being  reduced, 
gives  I^T  for  the  answer. 


1.  Multiply^  by  £/ 


.  14f. 


2.  Multiply^!  by  I|. 


3.  Multiply  ~-l  by  -f. 


4.  Multiply  ~|  by  ^|, 

,    ...      .        134 ,      20i- 

5.  Multiply  -  |  by  -— J. 


Am.  5flf|, 


6.  Multiply  — ^  by  — f . 

F  J      2J     J   22{- 

644 .        5J- 

7.  Multiply  — |  by  ^f 


8.  Multiply  ^p  by  -| 


COMPLEX   FKACTIONS.  225 

LESSON  XXXIII. 

257.    DIVISION    OF   COMPLEX   FRACTIONS. 
EXAMPLE. 


9  « 

Reduced  to  improper  fractions  gives  ~  by  -^r 

Placed  in  the  form  of  division  gives  (f-f-^)  by 
(t"^"V")'  Inverting  the  divisors  gives  (f  x^j)  by 
(f  x  A)-  The  last  complex  fraction  reduced  (|  x  ^T) 
being  the  divisor  of  the  given  sum,  invert  according 
to  the  rule,  in  which  case  all  fractions  of  the  dividend 

9          4         2         21 

are   inverted,  and  the  sum  is-—  x  —  x--x  —  . 

^1         13         5  o 

Cancelled  and  multiplied  gives  |^|'     Reducing  gives 


1.  Divide  jji  by  i|.  Ana. 

2.  Divide  ^  by  i|. 

3         yi" 

3.  Divide  1-  by  ^ 

4.  Divide      i-  by        . 


5.  Divide          by        . 


11.  Divide^  by  fi. 


226  INTERMEDIATE   AEITHMETIC. 

6.  DividegJ  bv  g. 

7.  Divide^  by  g|. 

8.  Divide?    by  ?.  Am. 


9.  Divide        by      . 

5  4 

10.  Divide  ||  by  it 

13  T 


12.  Divide  if  by  ^.  Ant. 


13.  Divide  by      . 

14.  Divide  ^|  by  "^. 

15.  Divide  |^  by  ^.  ^w».  23JJJ. 

3J  ^4: 

16.  Divide  g  by  ^|. 

17.  Divide  ff  by  i. 

T9"  3 

18.  Divide  ^  by  i.  Ana.  2j 

19.  Divide  i|  by  i  ^ns. 

!f         f 


USE   OF   BRACKETS.  227 


20.  Divide  ||  by  -|-.  Am. 


-  Divide       by 


22.  Divide       by      .  Am. 

15  ~8 


23.  Divide       by      .  jln*.  3-J  J  -J. 

T¥         » 


LESSON  XXXIV. 

238.  Brackets  are  used  for  the  purpose  of  en- 
closing a  set  of  numbers  which  must  be  considered 
separately  from  all  others  in  a  problem.  Thus: 
(l-f2  +  3)—  4  means  that  1,  2,  and  3  must  be  added 
together  before  the  figure  4  is  to  be  subtracted  from 
them. 

EXAMPLES. 

1.  Find  the  sum  of  (2  +  3+4)— (5)  =  9— 5  =  4. 

2.  Find  the  sum  of  (4— 2)X3^2  X  3=6. 

3.  Find  the  sum  of  (5—2)  X  3)— 4  =  (3  X  3)— 4  = 
9—4= 5. 

4.  Find  the  value  of  (i+f +J)— (J+f). 

Ans.  -ftf . 

5.  Find  the  value  of  (J+f +|)  X  (J-f  f ). 

Am.  2T5-gV 

6.  Find  the  value  of  (i  X  }) -5-  (f  X  i).      Am.  ||. 


228  INTERMEDIATE    ARITHMETIC. 

.  7.  Find  the  value  of  (J-4-£)  X  (f-H).        Ans.  |. 

8.  Find  the  value  of  (A  X|)+(|-i).  ^w«.  lif. 

9.  Find  the  value  of  (yV^-f ) — ($ — I)-     -^»*. 

10.  Find  the  value  of  (f— -J-)  -f-  (f+f).  -^ws.  T%. 

11.  Find  the  value  of  (-f-r-f)  X  (i+i).  -4w*.  H- 

12.  Find  the  value  of  (•£+}) — (44~4)'  Ans.  -|-|. 

13.  Find  the  value  of  (f — i)-f  (4 -?-{•)•  Ans.  |-|. 

14.  Find  the  value  of  (-J-H)— (J— i).  ^^^.  2f-|. 

15.  Find  the  value  of  (^  X  f  X  |)  —  J-.  ^^5.  ^3g-. 

16.  Find  the  value  of  (12—1)  X  (£  X  |).    ^ws.  ^J. 

1 7.  Find  the  value  of  (3+4)  x  (£—4).  ^-  6ft. 

18.  Find  the  value  of  (J— 4)  -7-(J  x  4).  ^Iwa.  12. 

19.  Find  the  value  of  (£  X  4)-r-  (f  X  f ).  Am.  yV 

20.  Find  the  value  of  (4+J)  X  (4-r-i).  ^ws.  f|. 

21.  Find  the  value  of  (f+|)-^(4^i).  -4ws. 

22.  Find  the  value  of  (ff -^ff)-H4  X  4  X  If). 

ute.  82. 


LESSON"   XXXV. 
23.  Find  the  product  of 

Ax 


MISCELLANEOUS  EXAMPLES. 

24.  What  is  the  sum  of 


ii±i±iL_W3 

-xr 


229 


.  420^. 


25.  Divide 


9| 


26.  Find  the  answer  for 


^_ 
- 


27.  Find  the  answer  for 


_ 
i 


/2m     20}     56}     8J     2|^ 


26 


8 


20 


230  INTERMEDIATE  ARITHMETIC. 


CHAPTER  XL 

LESSON  I. 

25O.  THE  EECIPKOCAL  of  any  number  is  unity 
(1)  divided  by  the  number. 

The  reciprocal  of  a  fraction  is  unity  divided  by 
the  fraction  which  is  the  same  as  inverting  the  terms 
of  the  fractions. 

EXAMPLES. 

The  reciprocals  of  2,  3,  4,  5,  etc.,  are  -J,  -J-,  J,  -J-,  etc. 
The  reciprocal^  of  £,  f,  f,  f,  are  f,  f,  f,  |. 
To  find  the  reciprocal  of  a  fraction  divide  1  by 
the  fraction  and  the  terms  will  be  inverted. 
Thus  1-7-f  =1  X  |  -|,  the  reciprocal  off. 

1.  What  are  the  reciprocals  of  2,  4,  6,  8,  10,  12, 
14,  1C,  18,  20,  22,  24,  26,  28,  30,  32,  34,  36,  38,  40? 

2.  What  are  the  reciprocals  of  £-,  f,  -f,  |,  f,  J,  |, 


3.  When  a  fraction  is  a  mixed  number,  reduce 
to  an  improper  fraction  and  invert  the  terms.  What 
are  the  reciprocals  of  1|,  2|,  3},  4f,  5f,  6-f,  7J,  8f, 


4.  What  is  the  reciprocal  of  }  of  f.     |-  x  |  =TV 
Inverted  gives  ^,  Ans. 

5.  What  are  the  reciprocals  of  \  of  4J?  f  of  |-  ?  1-} 

' 


DECIMALS.  231 


CHAPTER    XII. 

DECIMALS. 
LESSON    I. 

26O.  IF  an  apple  be  divided  into  10  equal  parts, 
each  part  would  be  one  tenth  of  the  whole  apple. 
Again,  if  one  of  the  tenths  be  divided  into  10  equal 
parts,  each  one  of  the  parts  is  called  one  hundredth 
of  the  apple. 

20 1.  The  above  expressions  are  written  two 
ways;  either  in  the  form  of  a  fraction  having  a 
denominator,  as  ^  T^  or  in  the  form  of  a  fraction 
whose  denominator  is  understood,  as  .1,  .01.  The 
last  expressions  are  called  decimal  fractions.  From 
the  above  it  will  be  seen  that  the  denominator  of  a 
decimal  fraction  is  10  or  any  number  of  10's. 

S6S.  Decimal  fractions  are  generally  written 
without  the  denominator,  the  period  or  decimal  point 
serving  to  show  where  the  numerator  begins.  All 
the  figures  to  the  right  of  the  decimal  point  make  the 
numerator,  those,  to  the  left  are  whole  numbers ; 
thus,  45  J  23  is  read,  45  whole  ones  and  123  thou- 
sands. Therefore  the  denominator  of  a  decimal 
fraction  is  1  with  as  many  O's  as  there  are  figures  to 
the  right  of  the  decimal  point.  Every  cipher  writ- 
ten between  the  decimal  figures  and  the  decimal 


232 


INTERMEDIATE   ARITHMETIC. 


point  decreases  the  value  ten  fold.  Thus  .4  is  4 
tenths,  .04  is  4  hundredths,  .004  is  4  thousandths, 
etc.,  etc.,  etc.  Ciphers  placed  to  the  right  of  any 
figure  after  the  decimal  point  do  not  change  .the 
value  of  the  fraction  ;  thus,  4  tenths,  40  hundredths, 
400  thousandths,  etc.,  etc.,  are  of  the  same  value. 

263.  The  first  place  to  the  right  of  the  decimal 
point  is  called  tenths,  2d,  hundredths,  3d,  thou- 
sandths, etc.,  etc. 


TABLE. 


7G543 


Whole  numbers. 


3456 


Decimals. 

A  mixed  decimal  number  is  a  whole  num- 
ber and  a  fraction  ;  as,  2.5,  two  and  five  tenths. 

A  complex  decimal  fraction  is  a  decimal  fraction 
and  a  common  fraction;  thus,  .2-J,  two  and  one  third 
tenths. 

NUMERATION.          * 

265.  To  read  decimal  numbers  : 

EULE. 

Head  the  decimal  figures  as  if  they  were  whole 
numbers,  and  add  the  name  of  the  right  liand  figure. 


DECIMALS.  233 

Thus,  read  .2046. 

The  number  of  places  is  4.  The  denominator  is 
then  ten  thousandths.  It  is  read  two  thousand  and 
forty-six  ten  thousandths. 

Read  the  following : 

.5  .8  .7  .3  .9  .4  .0  .2  .1  .10  .11  .12 
.13  .14  .16  .28  .92  .43  .105  .111  .246 
.384  .975  .287  .693  .108  .1012  .1001  .2003 
.3976  .1876  .1869  .42976  .40976  .40076 
.40006  .98764  .764921  .300010  .401010 
.870001  .287641. 

LESSON    II. 

NOTATION. 

266.  To  write  decimals. 

"Write  the  decimal  as  if  it  were  a  whole  number. 
Supply  the  vacant  spaces  with  ciphers,  and  be  care- 
ful to  have  as  many  figures  in  the  numerator  as 
there  are  O's  in  the  denominator.  Never  omit  the 
period  on  the  left  of  the  decimal  figures. 

EXAMPLE. 

£267.  Write  in  figures  one  hundred  and  one 
thousandths.  In  thousands  there  are  three  O's, 
there  must  be  three  figures  in  the  numerator.  One 
hundred  and  one  is  composed  of  three  figures.  Write 
one  hundred  and  one  as  if  it  was  a  whole  number, 
and  place  the  decimal  point  to  the  left  of  it.  Thus, 
.101 

1.  Write  the  following  :  five  tenths,  ten  hund- 


234 


INTERMEDIATE   ARITHMETIC. 


redths,  one  hundred  and  five  ten  thousandths,  one 
thousand  and  one  hundreds  of  thousandths.  Three 
hundred  thousand  and  ten  millionths.  Forty  thou- 
sand nine  hundred  and  seventy-six  one  hundred 
thousandths. 

LESSON  III. 

S68.   ADDITION   OF   DECIMALS. 
RULE. 

Write  the  numbers  under  each  other,  so  that  the 
figures  of  the  same  value  fall  under  each  other.  Add 
as  in  whole  numbers. 

EXAMPLES. 

1.  Add  2.461,  31.96  and  841.2581. 

Write  the  numbers  so  that  the 
figures  of  the  same  decimal  place  will 
fall  one  under  the  other.  Add  and 
point  off  as  many  figures,  beginning 
at  the  right,  as  there  are  decimal  fig- 
ures in  the  number  having  the  great- 
est number  of  decimals. 


OPERATION. 
2.461 

31.96 
841.2581 

875.6791 


2.  Add  1234.56,  123.456,  and  12.3456. 

Am.  1370.3616. 

3.  Add  2.34567,  234.567,  and  2345.67. 

Ans.  2582.58267. 

4.  Add  34567.8,  345.678,  and  345.678. 

Ans.  35259.156. 

5.  Add  45.6789,  45678.9,  and  45678.9. 

Ans.  91403.4789. 


ADDITION   OF  DECIMALS.  235 

G.  Add  50.7891,  567.891,  and  5078.91. 

Am.  0303.5901. 

7.  Add  6789.10,  678.910,  and  67.8910. 

Ans.  7535.901. 

8.  Add  789.11,  7891.1,  and  78911. 

Ans.  87591.21. 

9.  Add  89.101,  8.9101,  and  .89101. 

Ans.  98.90211. 

10.  Add  912.34,  912.34,  and  9.1234. 

Ans.  1833.8034. 

11.  Add  1.2345,  1234.5,  and  123.45. 

Ans.  1359.1845. 

12.  Add  1.3456,  .13456,  and  1.3456. 

Ans.  2.592576. 

13.  Add  5.4321,  543.21,  and  54.321. 

Ans.  002.9631. 

14.  Add  65.432,  654.32,  and  6543.2. 

Ans.  7202.952. 

15.  Add  7.0543,  7654.3,  and  765.43. 

Ans.  8427.3843. 

16.  Add  87.654,  876.54,  and  87654. 

Ans.  88618.194. 

LESSON  IV. 

269.    SUBTRACTION   OF   DECIMALS. 

RULE. 

Write  the  subtrahend  under  the  minuend,  so  that 
the  decimals  of  the  same  order  will  be  found  under 
each  other,  and  subtract  as  in  whole  numbers. 


236 


INTERMEDIATE   ARITHMETIC. 


EXAMPLES. 

1.  From  25.1304  take  8.24. 

Supply  the  vacant  spaces  with 
and  subtract  as  in  whole  numbers. 


O'j 


2.  From  102.03  take  12.345. 

3.  From  12.030  take  2.3045. 

4.  From  1230.4  take  234.05. 

5.  From  123.40  take  20.405. 

6.  From  23.040  take  12.040. 

7.  From  340.50  take  60.872. 

8.  From  4050.5  take  968.43. 

9.  From  60.704  take  12.3042. 

10.  From  123.45  take  102.0302. 

11.  From  6780.9  take  187.643. 

12.  From  901.011  take  24.680. 

13.  From  12.3042  take  1.8764. 

14.  From  8976.41  take  102.36. 

15.  From  8090.706  take  1809.2. 

16.  From  24.6809  take  2.8096. 

17.  From  123807.2  take  6.3082. 

18.  From  2380.70  take  60.302. 

19.  From  642.890  take  630.89. 

20.  From  389.7602  take  60.987. 


OPERATION. 

25.1304 
8.2400 


16.9904 

Ans,  89.685. 

Ans.  9.7255. 

Ans.  996.35. 

Ans.  102.995. 

Ans.  11.00. 

Ans.  279.628. 

Ans.  3082.07. 

Ans.  48.3998. 

Ans.  21.4198. 

Ans.  6593.257. 

Ans.  876.331. 

Ans.  10.4278. 

Ans.  8874.05. 

Ans.  6281.506. 

Ans.  21.8713. 

Ans.  123800.8918. 

Ans.  2320.398. 

Ans.  12.00. 

Ans.  328.7732. 


MULTIPLICATION    OF    DECIMALS. 


237 


LESSON  V. 

37O.    MULTIPLICATION    OF    DECIMALS 


Multiply  13.24  by  2.13. 

Multiply  as  in  whole  numbers. 
From  the  product  cut  off  as  many 
figures,  counting  from  right  to  left, 
as  there  are  decimal  figures  in  both 
multiplier  and  multiplicand.  The 
figures  to  the  left  of  the  decimal 
point  represent  whole  numbers  and 
those  to  the  right  represent  the  frac- 
tional part. 

EXAMPLES. 


OPERATION. 


28.20  12 


1.  Multiply  12.3  by  12. 

Ans.  147.6. 

2.  Multiply  12.34  by  12.3. 

Ans.  151.782. 

3.  Multiply  1.234  by  32.1. 

Ans.  39.6114. 

4.  Multiply  43.21  by  2.13. 

Ans.  92.0373. 

5.  Multiply  34.12  by  3.12. 

Ans.  106.4544. 

6.  Multiply  876.4  by  5.15. 

Ans.  4513.46. 

7.  Multiply  786.4  by  5.51. 

Ans.  4333.064. 

8.  Multiply  7.864  by  15.5. 

Ans.  12.1892. 

9.  Multiply  68.74  by  1.24. 

Ans.  85.2376. 

10.  Multiply  87.64  by  .214. 

Ans.  18.75496. 

11.  Multiply  21.87  by  .313. 

Ans.  6.84531. 

12.  Multiply  187.2  by  .331. 

Ans.  61.9632. 

13.  Multiply  18.76  by  .248. 

Ans.  4.65248. 

14.  Multiply  76.98  by  21.83. 

Ans.  1680.4734. 

15.  Multiply  187.64  by  28.96. 

Ans.  5434.0544. 

238 


INTERMEDIATE   ARITHMETIC. 


16.  Multiply  9898.98  by  .1111.  Ans.  1099.776678. 

17.  Multiply  8989.89  by  2.222.  Ans.  19975.55558. 

18.  Multiply  7643.924  by  121.212. 

Ans.  926535.315888. 


LESSOR  VI. 
19.  Multiply  .0121  by  4.1. 

S71.  Multiply  as  in  whole  num- 
bers. From  the  product  cut  off  as 
many  figures,  counting  from  right  to 
left,  as  there  are  decimals  in  both 
multiplier  and  multiplicand.  If  there 
is  not  as  many  figures  in  the  product 
as  decimals  in  both  multiplier  and 
multiplicand,  prefix  the  required  num- 
ber of  ciphers. 

EXAMPLES. 


OPERATION. 

.0121 
4.1 


.04961 


20.  Multiply  1.234  by  .02. 

Ans.  .02460. 

21.  Multiply  1.324  by  .021. 

Ans.  .027804. 

22.  Multiply  2.134  by  .0213. 

Ans.  .0454542. 

23.  Multiply  4.132  by  ,0287. 

Ans.  .1185884. 

24.  Multiply  3.142  by  .003. 

Ans.  .009426. 

25.  Multiply  4.312  by  .008. 

Ans.  034496. 

26.  Multiply  6.492  by  .0092. 

Ans.  .0597264. 

27.  Multiply  2.946  by  .0028. 

Ans.  .0082488. 

28.  Multiply  9.246  by  .0093. 

Ans.  .0859878. 

29.  Multiply  9.642  by  .087. 

Ans.  .838854. 

30.  Multiply  8.287  by  .021. 

Ans.  .174027. 

MULTIPLICATION    OP   DECIMALS. 


239 


31.  Multiply  2.878  by  .082.  Am.  .235996. 

32.  Multiply  8.782  by  .002.  Am.  .017564. 

33.  Multiply  2.788  by  .0003.  Am.  .0008364. 

34.  Multiply  9.673  by  .0121.  Am.  .1170433. 

35.  Multiply  7.693  by  .0128.  Am.  .0984704. 

36.  Multiply  8.428  by  .0212.  Am.  .1786736. 

37.  Multiply  8.367  by  .0218.  Ans.  .1824006. 

38.  Multiply  2.892  by  .0812.  Ans.  .2348304. 


LESSON  VII. 

272.  NOTE. — Multiplying  any  decimal  by  10,  100,  1000,  etc., 
removes  the  decimal  point  1,  2,  3,  etc.,  periods  to  the  left.  If 
there  be  not  enough  of  places,  annex  ciphers. 


39.  Multiply 

40.  Multiply 

41.  Multiply 

42.  Multiply 

43.  Multiply 

44.  Multiply 

45.  Multiply 

46.  Multiply 

47.  Multiply 

48.  Multiply 

49.  Multiply 

50.  Multiply 

51.  Multiply 

52.  Multiply 

53.  Multiply 


EXAMPLES. 

24.68  by  10. 
46.28  by  100. 
86.42  by  1000. 
24.86  by  10. 
98.743  by  100. 
34.789  by  100. 
4.3789  by  1000. 
74.389  by  100. 
8.7439  by  1000. 
8974.3  by  10. 
78.943  by  100. 
47.893  by  1000. 
218.726  by  10. 
281.721  by  100. 
821.217  bv  100. 


Ans.  246.8, 

Ans.  4628. 

Ans.  86420. 

Ans.  248.6. 

Ans.  9874.3. 

Ans.  3478.9. 

Ans.  4378.9. 

Ans.  7438.9. 

Ans.  8743.9. 

Ans.  89743. 

Ans.  7894.3. 

Ans.  47893. 

Ans.  2187.26. 

Ans.  2817.21. 

Ans.  82121.7. 


240 


INTERMEDIATE    ARITHMETIC. 


54.  Multiply  281.721  by  ICO.  Am.  28172.1. 

55.  Multiply  67.434  by  100.  Ans.  674.34. 
53.  Multiply  8972.3  by  100.  Ans.  897230. 

57.  Multiply  67.348  by  1000.  Ans.  67348. 

58.  Multiply  32.798  by  1000.  Ans.  32798. 

59.  Multiply  2972.43  by  10.  Ans.  29724.3. 

60.  Multiply  8.64398  by  100000.  Ans.  864398. 

61.  Multiply  9.87248  by  100000.  Ans.  987248. 

62.  Multiply  9.876432  by  1000000.  Ans.  9876432. 


LESSON  VIII. 


.    DIVISION   OF   DECIMALS. 


OPERATION. 

2.41)25.40(10.539  + 
24.1 

1300 
1205 


Divide  25.4  by  2.41. 

In  the  divisor  there  are 
2  decimal  figures ;  in  the 
dividend  but  1.  Make  the 
number  of  decimal  figures 
equal  in  both,  by  supplying 
the  deficiency  with  O's,  as  in 
the  example,  and  divide  as 
in  whole  numbers.  The 
quotient  thus  found  will  be 
the  whole  number.  When 
there  is  a  remainder,  annex 
a  cipher,  after  writing  the 
decimal  point  in  the  quo- 
tient to  the  right  of  the  whole  number,  and  continue 
the  division  as  far  as  you  think  proper. 


950 
723 

2270 
2169 

101 


DIVISION    OF    DECIMALS. 


241 


1.  Divide 

2.  Divide 

3.  Divide 

4.  Divide 

5.  Divide 

6.  Divide 

7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 

12.  Divide 

13.  Divide 

14.  Divide 

15.  Divide 


EXAMPLES. 

864.32  by  12.345. 
264.38  by  12.345. 
197.64  by  12.345. 
419.76  by  54.321. 
64.197  by  87.643. 
76.419  by  22.876. 
647.91  by  98.724. 
28.435  by  22.472. 
82.345  by  1.212. 
32.854  by  13.13. 
43.285  by  14.14. 
54.328  by  15.15. 
85.432  by  16.16. 
679.482  by  17.17. 
728.4365  by  28.28. 


Ans,  70.01  3J|f  j|. 
Ans.  21.415|||f. 

Ans.  16. 
Ans.  7.7 
Ans.  .7 

Ans.  8.340fffjf. 
Ans.  6.562gf  f-«. 


Ans.  67. 

Ans.  2.501 

Ans.  3.06  U  f}  f 

Ans.  3.586^-j. 

Ans.  6.286£f£. 

Ans.  39.515^-^. 

Ans.  2 


LESSON  IX. 
374.  Divide  .12  by  3.4. 

In  the  dividend  we  Lave 
two  decimals,  and  in  the  di- 
visor but  one.  Annex  an  0 
to  the  divisor.  It  now  be- 
comes 3.40.  The  decimal 
places  being  even,  we  now 
divide  as  in  whole  numbers. 
340  into  12,  0  times,  0  whole 
ones.  Annex  a  cipher,  mak- 
ing 120  tenths.  340  into  120  tenths,  0  tenths. 
21 


OPERATION. 

3.40). 1200(0.035  + 
1020 

1800 
1700 


100 


An- 


242 


INTERMEDIATE    ARITHMETIC. 


nex  another  cipher,  making  it  hundre-dths  (1200 
hundredths).  340  into  1200,  3  hundred  times 
and  180  hundredths  over.  Add  an  0  to  the  re- 
mainder, making  it  1800  thousandths.  340  into 
1800,  5  thousand  times  and  100  over,  etc.,  etc., 
etc. 


16.  Divide  2.46  by  11.11. 

Ans.  .221TYA. 

17.  Divide  4.62  by  22.22. 

Am.  .207^ff. 

18.  Divide  6.42  by  33.33. 

Ans.  .192Wf. 

O  O  O  0 

19.  Divide  4.26  by  44.44. 

Ans.  .095fff|-  . 

20.  Divide  6.24  by  55.55. 

Ans.  .112jft£. 

21.  Divide  9.'87  by  66.66. 

Ans.  .148J-IH. 

22.  Divide  3.48  by  7.777. 

Ans.  .044fff|. 

23.  Divide  9.82  by  88.88. 

Ans.  -11  Off!!. 

24.  Divide  7.64  by  9.999. 

Ans.  .764^\6/g. 

25.  Divide  9.81  by  10.111. 

Ans.  .970^/VVV 

26.  Divide  24.61  by  112.22. 

Ans.  .219-H-ff 

27.  Divide  8.97  by  12.333. 

Ans.  .727^\°TV 

28.  Divide  3.82  by  14.1414. 

Ans.  .270TyTVV 

29.  Divide  9.43  by  15.1617. 

Ans.  621f|ffff. 

30.  Divide  8.64  by  12.345. 

Ans.  ,699|ff. 

REDUCTION    OF    COMMON    FRACTIONS. 


243 


CHAPTER  XIII. 


LESSON  I. 

REDUCTION    OF    COMMON   FRACTIONS     TO    DE- 


CIMALS. 


GENERAL    RULE. 
Divide  the  numerator  by  the  denominator. 


1.  Reduce  -J  to  a  decimal  fraction. 
8   into    7    units,    0    times,    or   0 

units.  Annexing  an  0  to  the  7  makes 
it  tens.  8  into  70  tens,  8  tenths 
times.  8  times  8  are  64,  and  6  re- 
maining. Annexing  an  0  to  the  6, 
makes  it  60  hundredths.  8  into 
60,  7  times,  arid  4  hundredths  over. 
Annexing  an  0  to  the  4  makes  it  40 
thousandths.  8  into  40,  5  times, 
and  0  remaining.  If  the  decimal 
figure  repeats,  write  the  remainder 
in  the  form  of  a  fraction.  Thus, 

2.  Reduce  f  to  a  decimal  fraction. 
By  dividing,  we  find  2  remaining 

at  everylsubtraction.  Carry  the  deci- 
mal to  2  or  3  figures,  and  write  the  re- 
mainder jn  the  form  of  a  fraction. 
Before  dividing,  reduce  the  fractions 


to    their    lowest 
given. 


terms,    if   not    so 


OPERATION. 

8)70(0.875 
64 

60 
56 

40 
40 


OPERATION". 

3)20(0.66f 
18 

20 
18 


244  INTERMEDIATE   ARITHMETIC. 

3.  Reduce  |  to  a  decimal  fraction. 

Reduced  to  its  lowest  terms,  gives 

1- 

If  the  given  fraction  is  improper, 

divide  as  before. 

Thus : 

4.  Reduce  f  to  a  decimal  fraction. 

4  into  5,  once,  or  1  whole  one. 
Write  1,  and  place  the  point  to  the 
right  of  it.  To  the  remainder  annex 
an  0,  and  divide  as  before. 


OPERATION. 

4)30(0.75 

28 

20 
20 


OPERATION. 

4)5(1.25 
4 

10 

8 

20 


.  If  the  given  fraction  is  a  mixed  number: 
1st.  Reduce  to  an  improper  fraction,  and  divide. 

Or: 

2d.  Find  the  value  of  the  fractional  part,  and 
write  it  to  the  right  of  the  whole  number  with  the  deci- 
mal point  between  the  whole  number  and  the  decimal 
fraction. 

Thus : 


5.  Reduce  2|  to  a  decimal  fraction. 


11EDUCT1ON   OP   COMMON   FRACTIONS. 


1st.   Reduced  to   an   improper 
fraction  gives  ^.    Divide  as  above. 


Or: 


2d.  Find  the  value  of  -J 
by  the  general  rule.  By  the  oper- 
ation we  find  it  to  be  .75.  Write 
.75  to  the  right  of  the  whole  num- 
ber 2,  and  we  find  the  same  ans*wer 
as  in  the  first  case,  viz.,  2.75. 


OPERATION. 

4  )  11  (  2.75 
8 

30 

28 

20 
20 


OPERATION. 

4  )  30  (  0.75 

28 

20 


If  a  complex  fraction  is  given,  reduce  it  to 
a  simple  fraction  and  divide  as  before. 

279.    Before   dividing,   all   fractions  must   be 
reduced  to  their  simplest  forms. 

EXAMPLES. 

1.  Reduce  |-  to  a  decimal  fraction.  Ans.  0.5. 

2.  Reduce  -J-  to  a  decimal  fraction.  Ans.  0.333^. 

3.  Reduce  |-  to  a  decimal  fraction.  Ans.  0.25. 

4.  Reduce  -J-  to  a  decimal  fraction.  Ans.  0.2. 

5.  Reduce  ^-to  a  decimal  fraction.  Ans.  O.lGGf. 

6.  Reduce  \  to  a  decimal  fraction.  Ans.  0.142-|. 

7.  Reduce  4-  to  a  decimal  fraction.  Ans.  0.125. 


246 


INTERMEDIATE   AEITHMETIC. 


8.  Reduce  J  to  a  decimal  fraction.    Am. 

9.  Reduce  T^  to  a  decimal  fraction.        Ans.  0.1. 

10.  Reduce  §  to  a  decimal  fraction.    Ans.  0.666|. 

11.  Reduce  }  to  a  decimal  fraction.        Ans.  0.75- 

12.  Reduce  %  to  a  decimal  fraction.         Ans.  0.8. 

13.  Reduce  £  to  a  decimal  fraction.    Ans.  0.833  J. 

14.  Reduce  -f  to  a  decimal  fraction.   Ans.  0.857^. 

15.  Reduce  f  to  a  decimal  fraction.      Ans.  0.875. 
1C.  Reduce  |  to  a  decimal  fraction.    Ans.  0.888JJ-. 

17.  Reduce  y^  to  a  decimal  fraction.       Ans.  0.9. 

18.  Reduce  £  +  -J-  to  a  decimal  fraction. 

Ans.  0.833}. 
-J-4-f  to  a  decimal  fraction. 


19.  Reduce 

20.  Reduce  •}  +  J  to  a  decimal  fraction. 

21.  Reduce 

22.  Reduce 


1.166f. 
1.083-}. 


to  a  decimal  fraction. 

Ans.  1.025. 
to  a  decimal  fraction. 

-4/zs.  .01111. 
23.  Reduce  -j-  —  T4^-  to  a  decimal  fraction. 


-J- 


24.  Reduce  f  —  ^  to  a  decimal  fraction.      ^4.^.9.  .5. 

25.  Reduce  f  —  j1^  to  a  decimal  fraction.  Ans.  .75. 

26.  Reduce  J  X  f  to  a  decimal  fraction. 

Ans.  .333?,, 

27.  Reduce  ^X  f  to  a  decimal  fraction.  Ans.  .25. 

28.  Reduce  ^  X  tV  to  a  decimal  fraction. 

Ans.  .1125. 

29.  Reduce  y^X  ^  to  a  decimal  fraction. 

Ans.  .0727I3J, 


REDUCTION   OF   DECIMALS.  247 

30.  Reduce  -f%  X  f-J  to  a  decimal  fraction. 

Ans.  .066f. 

31.  Reduce  J-hf  to  a  decimal  fraction.   Ans.  .75. 

32.  Reduce  ^-f-  J-  to  a  decimal  fraction. 

-4w*.  444*. 

33.  Reduce  -£-T-§  to  a  decimal  fraction,    ^im  .32. 

34.  Reduce  -\-  f-f  to  a  depimal  fraction. 

yl?is.  .1875. 

35.  Reduce  (-J-  +  J)  X  (J  —  £)  to  a  decimal  fraction. 


36.  Reduce  (-J--i-i)  +  (i-s-^r)  to  a  decimal  fraction. 

Ans.  2.236  >-. 

37.  Reduce  (-f")~*~i  to  a  ^cc™al  fraction. 

Ans.  1.92. 

38.  Reduce  (-§-^)  +  J  to  a  decimal  fraction. 

Ans.  .819f. 

LESSOR  II. 

28Oa  REDUCTION  OF  DECIMALS  TO  COMMON  FRACTIONS. 

KULE. 

Write  the  figures  of  the  given  number  for  the  nu- 
merator. 

Write  the  figure  1  followed  by  as  many  O's  as 
there  are  decimal  figures  in  the  given  number  for  the 
denominator. 

Reduce  the  fraction  thus  formed  to  its  lowest 
terms. 


248 


INTERMEDIATE   ARITHMETIC. 


EXAMPLE. 

Reduce  .75  to  a  common  fraction. 

In  this  sum  there  are  2  decimal  OPEEATIOX. 

figures.      Write   the   given   number,  JUL— 3. 
75,  for  the  numerator,  and  for   the 
denominator  write  1  followed  by  two  O's.     Reduc- 
ing the  fraction  thus  formed  gives  f  for  the  answer. 

1.  Reduce  .5  to  a  simple  fraction.  Am.  J. 

2.  Reduce  .05  to  a  simple  fraction.  Am.  -fa. 

3.  Reduce  .2  to  a  simple  fraction.  Ans.  -J-. 

4.  Reduce  .02  to  a  simple  fraction.  Ans.  -f^. 

5.  Reduce  .25  to  a  simple  fraction.  Ans.  J-. 

6.  Reduce  .025  to  a  simple  fraction.  Ans.  -fa. 

7.  Reduce  .175  to  a  simple  fraction.  Ans.  -fa. 

8.  Reduce  .0175  to  a  simple  fraction.  Ans.  -gfa. 

9.  Reduce  .24  to  a  simple  fraction.  Ans.  ^5-. 

10.  Reduce  .024  to  a  simple  fraction.  Ans.  yf-^. 

11.  Reduce  .85  to  a  simple  fraction.  Ans.  -J^. 

12.  Reduce  .085  to  a  simple  fraction.  Ans.  -£fa. 

13.  Reduce  0.125  to  a  simple  fraction.  Ans.  -J. 

14.  Reduce  .0125  to  a  simple  fraction.  Ans.  -gV 

15.  Reduce  0.1  to  a  simple  fraction.  Ans.-f^. 

16.  Reduce  .01  to  a  simple  fraction.  Ans.  TJ-y-. 

17.  Reduce  .75  to  a  simple  fraction.  Ans.  J. 

18.  Reduce  .075  to  a  simple  fraction.  Ans.  •£$. 

19.  Reduce  .8  to  a  simple  fraction.  Ans.  -f. 

20.  Reduce  .08  to  a  simple  fraction.  Ans.  -£%. 

21.  Reduce  .875  to  a  simple  fraction.  Ans.  -J. 

22.  Reduce  .0875  to  a  simple  fraction.  Ans.  -fo. 


INDUCTION    OF   DECIMALS.  249 

23.  Reduce  .9  to  a  simple  fraction.  Ans.  ^. 

24.  Reduce  .09  to  a  simple  fraction.  Ans.  ^fa. 

25.  Reduce  .1125  to  a  simple  fraction.  Ans.  -fc. 

26.  Reduce  .01125  to  a  simple  fraction.  Ans.  -j;,,. 

27.  Reduce  0.32  to  a  simple  fraction.  Ans.  -/j. 

28.  Reduce  .032  to  a  simple  fraction.  Ans.  yf-y. 

29.  Reduce  .1875  to  a  simple  fraction.  Ans.  -fa. 

30.  Reduce  .01875  to  a  simple  fraction.  Ans.  1  •!„. 

31.  Reduce  .92  to  a  simple  fraction.  Ans.  |5. 

32.  Reduce  .092  to  a  simple  fraction.  Ans. 

33.  Reduce  .6  to  a  simple  fraction.  Ans.  |. 

34.  Reduce  .66  to  a  simple  fraction.  Ans.  •?;";. 

35.  Reduce  .06  to  a  simple  fraction.  Ans.  -/$. 

36.  Reduce  .066  to  a  simple  fraction.  Ans.  -f^. 

37.  Reduce  .486  to  a  simple  fraction.  Ans.  -?••;";. 

38.  Reduce  .648  to  a  simple  fraction.  Ans. 

39.  Reduce  .325  to  a  simple  fraction.  Ans.  Jf . 

LESSON"  III. 

28 1 .  When  there  is  an  irreducible  fraction  at  the 
end  of  the  decimal. 

EXAMPLE. 

Reduce  .22f  to  a  simple  fraction. 

Multiply  the  decimal  by  the  denominator,  and 
to  the  result  add  the  numerator,  as  in  reduction  of 
mixed  numbers  to  improper  fractions.  Under  the 


250 


INTERMEDIATE   ARITHMETIC. 


result  write  the  denominator  followed  by  as  many 
O's  as  there  are  decimal  figures  in  the  given  number. 

000 

(9X22)4-2  =  200.     ^.     Reduced  gives  f. 
Reduce  the  fraction  to  its  lowest  terms. 


EULE. 

Iteduce  the  decimal  as  in  mixed  numbers,  and  for 
the  denominator,  write  the  denominator  of  the  frac- 
tional part  followed  by  as  many  O's  as  there  are  deci- 
mal figures  in  the  given  number. 

Reduce  the  fraction  thus  formed  to  its  lowest 
terms. 


EXAMPLES. 


1.  Reduce 

2.  Reduce 

3.  Reduce 

4.  Reduce 

5.  Reduce 

6.  Reduce 

7.  Reduce 

8.  Reduce 

9.  Reduce 

10.  Reduce 

11.  Reduce 

12.  Reduce 

13.  Reduce 

14.  Reduce 


.33^  t6  simple  fractions. 

.166|  to  simple  fractions. 

.142-f  to  simple  fractions. 

.111&  to  simple  fractions. 

.66f  to  simple  fractions. 

.833^  to  simple  fractions. 

.857-}-  to.  simple  fractions. 

.888f  to  simple  fractions. 

.106f  to  a  common  fraction.  Ans.  - 

.083^  to  a  common  fraction.  Ans.  - 

.0111^  to  a  common  fraction. 

Ans.  -j 

.155f  to  a  common  fraction.  Ans. 
.33|  to  a  common  fraction.  Ans.  J 
.25  3^  to  a  common  fraction.  Ans.  % 


Ans. 
Ans. 
Ans.  \. 
Ans.  |. 
Ans.%  . 
Ans.  f  . 
Ans.  f. 
Ans.  f  . 


REDUCTION    OF    DECIMALS.  251 

15.  Reduce  .82^  to  a  common  fraction.  Am.  f-fj. 

16.  Reduce  .0727T\  to  a  common  fraction. 

An.  . 

17.  Reduce  .OGOGf  to  a  common  fraction. 

Ans.  TJ  ,1,  ,-,  . 

18.  Reduce  .44$  to  a  common  fraction.        Ans.$. 

19.  Reduce  .925f-f-  to  a  common  fraction.  Ans.  -%%. 

20.  Reduce  .236^  to  a  common  fraction.  Ans.  -£}. 

21.  Reduce  .819$  to  a  common  fraction.  Ans.  f£. 

22.  Reduce  .047^  to  a  common  fraction. 


23.  Reduce  ,28-J  to  a  common  fraction.  Ans.  |-  J-J-. 

24.  Reduce  .33|  to  a  common  fraction    Ans.  -^. 

25.  Reduce  .01|  to  a  common  fraction.     Ans.  -fa. 
20.  Reduce  .46-i-f  to  a  common  fraction.  Ans.  -^j^. 

27.  Reduce  .28^  to  a  common  fraction.       Ans.  f. 

28.  Reduce  .28^1T  to  a  common  fraction. 

Ans.  ^VoV 

29.  Reduce  .33f  to  a  common  fraction.  Ans.  |  •]  |}. 

30.  Reduce  .48^  to  a  common  fraction.     Ans.  |-j}. 

31.  Reduce  .26^-  to  a  common  fraction.  Ans.  -^\. 

32.  Reduce  .18f  to  a  common  fraction.     Ans.  ^. 

33.  Reduce  .16^  to  a  common  fraction.  Ans.  -^. 

34.  Reduce  .283^  to  a  common  fraction.  Ans.  ^. 

35.  Reduce  .24f  to  a  common  fraction.    Ans.  -£/%- 

36.  Reduce  ,2-J  to  a  common  fraction.       Ans.  J^. 

37.  Reduce  .1    to  a  common  fraction.       Ans.      . 


252 


INTERMEDIATE  ARITHMETIC. 


CHAPTER  XIII. 
REPETMDS. 

LESSON    I. 

S82.  IT  very  often  happens,  when  reducing  a 
common  fraction  to  a  decimal,  that  the  same  figures 
will  repeat  in  the  quotient  as  long  as  the  division  is 
carried  on.  In  order  to  show  what  figures  repeat,  a 
dot  (  . )  is  placed  over  them.  In  some  instances, 
several  sets  of  figures  repeat.  The  dot  is  placed  on 
the  first  and  last;  thus,  243243243.  In  a  quotient 
like  the  one  given,  but  3  figures  are  used,  because 
the  other  figures  of  the  dividend  are  the  same,  since 
they  are  but  a  repetition  of  the  numbers  between  the 
dots. 

283,  Whenever  a  dot  is  placed  over  a  decimal 
figure,  or  over  two  figures  between  which  there  are 
other  figures,  it  shows  that  the  figure  is  repeated 
indefinitely.     In  the  second  case  it  shows  that  the 
figures  over  which  the  dot  is  placed,  together  with 
the  figures  between  them,  repeat   in  the   order  in 
which  they  are  written.     Sometimes  a  number  like 
the  following  occurs  :  248.     It  shows  that  the  figure 
8  alone  repeats. 

284.  All  figures  between  the  dots  and  under 
the  dot  repeat,  and  none  others.     In  this  example 


EEPETENDS.  253 

(.3),  the  figure  3  repeats.  In  this  example,  246, 
the  number  240  repeats,  in  the  order  in  which  the 
figures  are  written. 

In  such  an  expression  as  the  following,  .2689,  the 
figures  2  and  6  do  not  repeat.  The  8  and  0  repeat 
in  the  order  in  which  they  are  written. 

Which  of  the  following  figures  repeat: 

2,  .3,  .6,  .8,  .7,  .5,  .28,  .49,  .67,  .37, 
.443*,  .684,  .772,  .846,  .2406,  .8927,  .2248, 
.3976,  .286,  .874,  .982,  .647,  .241*. 


LESSON  II. 

S8o.  When  the  decimal  repeats  and  that  the 
given  number  has  not  the  fractional  part  at  the  end 
of  the  decimal  number.  Thus  : 

Reduce  .46  to  a  simple  fraction. 


Write  the  number  46  for  the  nu- 


OPERATION. 


merator,   and    for   the    denominator 

n,          ..  ,     .  44     Ans. 

write  as  many  9s  as  there  are  deci- 

mal figures  in  the  given  number.     Reduce  the  frac- 
tion thus  formed  to  its  lowest  terms. 


EXAMPLES. 

1.  Reduce  .3  to  a  common  fraction.          Ans. 

2.  Reduce  .03,  to  a  common  fraction.  Ans. 

3.  Reduce  .47  to  a  common  fraction.  Ans. 

22 


254 


INTERMEDIATE    ARITHMETIC. 


4.  Reduce  .24  to  a  common  fraction. 

5.  Reduce  .36  to  a  common  fraction. 

6.  Reduce  .82  to  a  common  fraction. 

7.  Reduce  .34  to  a  common  fraction. 

8.  Reduce  .29  to  a  common  fraction. 

9.  Reduce  .27  to  a  common  fraction. 

10.  Reduce  .66  to  a  common  fraction. 

11.  Reduce  .is  to  a  common  fraction. 

12.  Reduce  .36  to  a  common  fraction. 

13.  Reduce  .63  to  a  common  fraction. 

14.  Reduce  .45  to  a  common  fraction. 

15.  Reduce  .54  to  a  common  fraction. 

16.  Reduce  .9  to  a  common  fraction. 

17.  Reduce  .si  to  a  common  fraction. 

18.  Reduce  .360  to  a  common  fraction. 

19.  Reduce  .12  to  a  common  fraction. 

20.  Reduce  .180  to  a  common  fraction. 


Am.  -j^. 

Ans.  T4T. 

Ans.  |-f 

Ans.  ||." 

Ans.  -jj-jj-. 

Ans.  T3T. 
Ans.  f. 

Ans.  T2T. 

Ans.  T4T. 

Ans.  T7T. 

Ans.  T5T. 

Ans.  T6T. 
Ans.  1. 

^4m.  T°T. 
Ans. 

Ans. 
Ans.  -f 


LESSOR  III. 

S8G.  When  there  is  a  finite  part  before  the  rep- 
etend.  Thus : 

Reduce  .46  to  a  simple  fraction. 

Make  a  mixed  number  of  it,  with  9  for  the  de- 
nominator of  the  repetend;  thus,  4{j-.  Divide  4-g-  by 
1  followed  by  as  many  O's  as  there  are  decimal  figures 


KEPETENDS.  255 

in  the  given  number,  after  having  reduced  it  to  a 
mixed  number,  which  would  be  in   this  case   but 

4*5. 
one  0.    The  fraction  would  now  appear  thus  :  --A 

or  4-}-  -i-  10.     Reduced  to  its  simplest  form  would  give 
)  f!  for  the  required  fraction. 


PEG  or. 
1.  Reduce  |^  to  a  decimal. 

45  )  210  (  0.466 
180 

300 
270 

300 
270 

30 

EXAMPLES. 

1.  Reduce  .24  to  a  common  fraction.  Am.  J£. 

2.  Reduce  .86  to  a  common  fraction.  Am.  -J-|. 

3.  Reduce  .38  to  a  common  fraction.  Ans.  T\. 

4.  Reduce  .73  to  a  common  fraction.  Ans.  ^\. 

5.  Reduce  .26  to  a  common  fraction.  Ans.  -fo. 

6.  Reduce  .62  to  a  common  fraction.  Ans.  j-f. 

7.  Reduce  .81  to  a  common  fraction.  Ans.  -£f. 

8.  Reduce  .97  to  a  common  fraction.  Ans.  £f. 

9.  Reduce  .36  to  a  common  fraction.  Am. 


256  INTERMEDIATE   ARITHMETIC. 

10.  Reduce  .28  to  a  common  fraction.  Ans.  j-f. 

11.  Reduce  .864  to  a  common  fraction.  Ans.  |f|.- 

12.  Reduce  .28  to  a  common  fraction.  Ans,  J-|. 

13.  Reduce  .82  to  a  common  fraction.  Ans.  f-J. 

14.  Reduce  .762  to  a  common  fraction.  Ans.  444-. 

i  y  o 

15.  Reduce  .9873  to  a  common  fraction.  Ans.  |~|-f. 

16.  Reduce  .876  to  a  common  fraction.   Ans.  Mi. 

4  J  D 

17.  Reduce  .283  to  a  common  fraction.     Ans.  -J-J. 

18.  Reduce  .246  to  a  common  fraction.  Ans.  -ffom 

19.  Reduce  .61  to  a  common  fraction.       Ans.  ^-. 

20.  Reduce  .861  to  a  common  fraction.     Ans.  - 

21.  Reduce  .248  to  a  common  fraction.   Ans.  -/- 

22.  Reduce  .726  to  a  common  fraction.    Ans.  ^-jf-jf. 

23.  Reduce  .278  to  a  common  fraction.  Ans.  |-jj-j-. 

24.  Reduce  .286  to  a  common  fraction.  Ans.  T4/0. 

25.  Reduce  .928  to  a  common  fraction.  Ans.  f|-f. 

26.  Reduce  .874  to  a  common  fraction.    Ans.  f  f-f. 

27.  Reduce  .639  to  a  common  fraction.    Ans.  -fJ-J-. 

28.  Reduce  .2876  to  a  common  fraction. 

Ans.  $£fo. 

29.  Reduce  .249  to  a  common  fraction.  Ans.  f  f -J-. 

30.  Reduce  .876  to  a  common  fraction.  Ans.  |ff. 

31.  Reduce  .881  to  a  common  fraction.  Ans.  ||f. 

32.  Reduce  .896  to  a  common  fraction.  Ans.  |-jj-g-. 

33.  Reduce  .284  to  a  common  fraction.  Ans.  -. 


MISCELLANEOUS    EXAMPLES.  257 

34.  Reduce  .38:)  to  a  common  fraction.     Ans.  f~§. 

35.  Reduce  .0491  to  a  common  fraction. 


30.  Reduce  .043  to  a  common  fraction.   Ans.  •£•]•;. 

37.  Reduce  .080  to  a  common  fraction.   Ans.  -{'J-j. 

38.  Reduce  .2833  to  a  common  fraction.   Ans.  -J-J. 

39.  Reduce  .8000  to  a  common  fraction.  Ans.      . 


PROBLEMS    INVOLVING   PRECEDING    HULES    AND 
PRINCIPLES. 

LESSON   IV. 

287.  1.  I  bought  250000  bricks,  at  12$  dollars 
per  thousand.  I  retailed  J-  of  them  to  I.  Smith  at  14 
dollars  per  thousand,  ^  of  the  remainder  to  J.  Jones 
at  15  dollars  per  thousand,  and  the  balance  to  Robert 
Peters  at  12  dollars  per  thousand.  Did  I  gain  or 
lose,  and  how  much  ?  Ans.  Lost  $78.75. 

2.  If  a  horse  eat  J-  bushel  of  oats  in   one  day, 
•J-  in  another,  and  TV  in  another,  how  many  bushels 
did  he  eat  ?  Ans.  -fy  bushels. 

3.  Sam.  Browne  bought  two  hhds.  of  sugar,  each 
weighing  2000  pounds,  at  4  cents  a  pound.     How 
much  profit  would  he  make  by  selling  it  at  4^  cents 
per  pound  ?  Ans.  $20. 

22* 


258  1XTEKMEDIATE   ARITHMETIC. 

4.  A  man  bought   two   pieces   of  cotton,   each 
piece   containing   82J-  yards.     He   sold   24|  yards. 
How  many  yards  had  he  left,  and  what  was  it  worth 
at  16  cents  a  yard  ?  Ans.  $22.38 

5.  A  grocer  bought  2  boxes  soap,  at  8  dollars 
apiece ;  2  bbls.  molasses,  40  gallons  each,  at  1  dollar 
per  gallon.     How  much  change  did  he  receive  from 
the  merchant  in  exchange  for  a  100  dollar  bill  ? 

Ans.  $4 

6.  A  merchant  bought  two  pieces  of  cloth,  20^ 
yards  each,  and  divided  it  between  8  persons.     How 
many  yards  did  each  one  receive?         Ans.  5^  yds. 

I.  From  %  of  a  dollar  subtract  -J-  of  a  dollar. 

Ans.  37^  cents. 

8.  I  buy  sugar  for  £  of  a  dollar,  and  I  sell  it  for  J 
of  a  dollar.     What  is  my  profit  ?  Ans.  12%  cts. 

9.  What  is  the  difference  between  -J  of  a  dollar 
and  Jg-  of  a  dollar  ?  Ans.  6  J  cts. 

10.  From  250  acres  of  land  I  gave  24|  acres. 
How  many  had  I  left  ?  Ans.  225|  A. 

II.  Divide  12$  dollars  among  100  boys. 

Ans.  12^  cts. 

12.  Ninety   cords   wood   cost    me    270   dollars- 
How  much  is  each  cord  worth  ?  Ans.  $3. 

13.  I  bought  90  cords  of  wood  at  $4^  a  cord. 
What  had  I  to  pay  ?  Ans.  $405. 

14.  A  teacher   bought  3  dozen  spellers  at  12£ 
cents  a  piece.     What  did  he  have  to  pay  ? 

Ans.  $4,50 


MISCELLANEOUS   EXAMPLKS.  259 

15.  A  merchant  bought  24  cords  of  wood  at 
cord,  and  sold  -?,  of  it  at  9  dollars  a  cord.     For  the 
balance  he  got  04  dollars.     How  much  did  he  sell 
the  balance  per  cord  ?  Ans.  $4. 

16.  I  bought  a  cask  of  wine  containing  50  gal- 
lons, at  1  dollar  a  gallon.     I  sold  it  to  a  grocer  for 
150  dollars.     How  much  per  gallon  did  the  grocer 
pay  for  it  ?  Ans.  $3. 

17.  A  train  has  to  travel  240  miles  from  one  sta- 
tion to  another.     How  long  will  it  take  the  train  to 
go  that  distance   if  it  should  travel   3d}    miles  an 
hour?  Ans.  7  hrs.  52  min. 

18.  If  I  sell  24000  bricks  at  12  dollars  a  thous- 
and, and  buy  the  same  number  of  bricks  at  $1.20 
per  hundred,  how  much  do  I  gain  ?  Ans. 

19.  A  boy,  after    losing  £  of   his    kite-string, 
bought  20  yards  of  cord,  and  added  it  to  wliat  he 
had  left.     He  now  finds  that  he  has  but  |-  of  what 
he  had  at  first.     How  many  feet  of  string  had  he  at 
first  ?  Ans.  240  yards. 

20.  I  own  f  of  a  ship  valued  at  $77,000.     If  I 
sell  j-  of  my  share,  how  much  will  I  receive,  and  how 
much  will  I  still  own  ? 

Ans.  Sold  $19,250.       Own  $6,416.66|. 

21.  A  and  B  are  150  miles    apart.     A  travels  2 
miles  an  hour,  and  B  travels  4  miles  an  hour.    When 
will  they  meet,  and  how  many  miles  will  each  one 
have  travelled?  Ans.  A  50  miles,  B  100  miles. 

22.  A  merchant   begins   business  with   $20,000. 
The  first  year  his  capital  increased  £  the  original 


260  INTERMEDIATE   AKITHMETTC. 

amount ;  the  second  year  he  lost  $2500.     How  much 
has  he  left  ?  Ans.  $27500. 

23.  Memphis  is  about  880  miles  from  New  Or- 
leans by  rail.     If  a  train  was  24  hours  in  making  that 
distance,  how  many  miles  an  hour  would  that  be  ? 

Ans.  36f  miles. 

24.  A  boy  gave  %  of  his  money  to  a  beggar,  £  for 
a  grammar,  and  \  for  a  reader,  and  now  he  has  20 
cents  left.     How  much  had  he  at  first  ?     Ans.  $1.60. 

25.  A  pole    is  |-  of  its  length  in  the  mud,  £  in 
water,  and  40  feet  above  the  water.     How  long  is 
the  pole?  Ans.  160  feet. 

26.  From  one  dollar  take  one  mill. 

Ans.  .999  mills. 

27.  What  will  2  hhds.  cost,  each  hogshead  weigh- 
ing 460  pounds  at  10£  cents  a  pound.      Ans.' $96.60. 

28.  I  bought  20  hhds.  wine,  63  gallons  each,  at 
22£  cents  per  gallon.     How  much  had  I  to  pay  ? 

Ans.  $280.35. 

29.  I  employ  14  men  for  10J-  days,  for  $2.25  per 
day.     How  much  is  my  expense  ?          Ans.  $330.75. 

30.  I  sold  to  Johnson  &  Co.  720  loads  of  brick ; 
each   load  contained  500  "bricks.     How  much  had 
Johnson  &  Co.  to  pay  at  $14.50  per  thousand  ? 

Ans.  $5220. 

31.  When   building  my  stables,  I  required,  to 
cover  them,  600  bundles  of  shingles ;   each  bundle 
contained  250.     How  many  shingles  did  I  buy,  and 
what  had  I  to  pay  for  them  at  $3.25  per  thousand  ? 

Ans.  $487.50. 


MISCELLANEOUS   EXAMPLES.  261 

32.  A  mason   requires  350000  bricks  to  build  a 
wall.     He   receives   600  loads  of  500  bricks  each. 
How  many  more  does  he  want,  and  what  would  he 
have  to  pay  for  them  at  $13  a  thousand  ?  Ans.  $650. 

33.  A  dressmaker  bought  24  yards  of  silk  at  33^ 
cents  a  yard.   How  much  had  she  to  pay  ?    Ans.  $8. 

34.  A  boy  after  losing  £  of  his  marbles  at  play, 
finds  that  he  has  20  left.     How  many  had  he  at 
first  ?  Ans.  40  marbles. 

35.  After   losing  £  of  his   fortune,  a   man   has 
$25000  left.     How  much  had  he  at  first  ? 

Ans.  $33333.33^. 

36.  Seven  eighths  of  the  steamer  Pocahontas  is 
valued  at  $70000.     James  Brown  owns  ^  of  her,  and 
he  sells  £  of  f  of  his  share  to  Smith.     How  much 
does  Smith  own  ?     How  much  has  Brown  left  ? 

Ans.  $17777.77f 

37.  If  in  a  school  ^  of  the  boys  study  grammai^ 
^  of  the  balance  study  arithmetic,  and  the  remainder 
study  history,  what  part  study  history  ?          Ans.  •£. 

38.  A  and  B  own   280  sheep,  worth  $5  apiece. 
A  owns  |-  and  B  the  remainder.     How  many  sheep 
does  each  one  own,   and  how  much  is  each  one's 
share  valued  at  ?  Ans.  A,  70  ;  B,  210. 

39.  A   man    begins    business   with    $1000    and 
doubles  his  capital  every  year  for  4  years.     How 
much  capital  has  he  at  the  end  of  the  fourth  year? 

Ans.  $16000. 

40.  A  can  do  a  piece  of  work  in  2  days  and  B 
can  do  it  in  3  days.     How  long  will  it  take  both  of 
them  to  do  it,  working  together?  Ans.  1|  day. 


262 


INTERMEDIATE   ARITHMETIC. 


41.  After  spending  |  of  my  income  every  year,  I 
find  that  I  save  $1200.     What  is  my  income  ? 

Am.  81500. 

42.  A  man  divides  $20000  between  his  4  sons  as 
follows :  A  gets  \  of  it,  B  gets  \  of  the  remainder, 
C  gets  £  of  what  is  now  left,  and  D  gets  the  balance. 
What  is  the  share  of  each  ? 

Ans.  A,  $5000;  B,  $3750;  C,  $2812.50; 
D,  $8437.50. 


CHAPTER   XIV. 

DENOMINATE  NUMBERS. 
LESSON  I. 

388.  A  compound  or  denominate  number  is  a 
collection  of  units  of  several  denominations;  thus, 
£2  3s.  4d.  2  far.  is  a  compound  number. 

38®.  A  compound  or  denominate  number  may 
be  changed  to  a  different  denomination  without 
altering  its  value. 

SOO.  There  are  two  ways  of  changing  the  denomi- 
nation, called  reduction  ascending  and  reduction  de- 
scending. 

3i$l.  Reduction  ascending  is  the  process  of 
reducing  from  a  lower  to  a  higher  denomination,  as 
pence  to  pounds,  etc. 

This  operation  is  performed  by  division. 


ENGLISH   MONEY.   .  203 

393.  Reduction  descending  is  the  process  of 
reducing  from  a  higher  to  a  lower  denomination,  as 
pounds  to  farthings. 

This  operation  is  performed  by  multiplication. 

ENGLISH    MONEY. 

393.  English  money  is  the  currency  of  England 
and  her  dependencies. 

394.  It  is  used  in  buying  and  selling  in  some  of 
the  Middle   and   Eastern   States,  but  it  is  almost 
obsolete. 


The  following  is  the 


TABLE. 

4  farthings  (far.)  make  1  penny,    abb.  d. 
12  pence  "      1  shilling,     "     s. 

20  shillings  "      1  pound,       "     £ 

21  shillings  u      1  guinea,      "     G. 

5  shillings  "      1  crown,      "     Cr. 

1  £     is  worth  $4.84  in  U.  S.  money. 

1  s.  "  0.24£  "  " 

1  d.  "  0.02001+  "  " 

1  far.         "          0.005004+          "  " 

EXERCISES. 

395.  1.  In  £4  how  many  farthings  ? 

OPEEATION. 

4  x  12  x  20  x  4  =  3840  farthings. 

In  such  examples  as  the  above,  begin  at  farthings, 
and  multiply  by  all  the  figures  until  you  arrive  at 


264  INTERMEDIATE    ARITHMETIC. 

the  required  point.  In  the  above  example  it  is 
required  to  go  from  farthings  to  pounds,  then  mul- 
tiply all  the  figures  from  farthings  to  pounds,  and 
since  the  question  is  £4,  you  will  require  4  times  as 
many  as  are  found  in  the  table.  If  the  question  was 
to  reduce  from  pence  to  pounds,  we  would  begin  at 
pence  and  multiply  until  we  came  to  pounds ;  thus, 
begin  at  pence,  12  pence  x  20  shillings  =  £1. 

This  remark  applies  to  all  of  the  tables  of  denom- 
inate numbers,  therefore  it  is  useless  'to  repeat  it 
again. 

2.  In    1    pound  how   many   farthings  ?     2  ?     3  ? 
4?     5?     6?     7?     9?     10?     11?     13?     14? 

3.  In  1  shilling  how  many  farthings  ?     2  ?     3  ? 
4?     5?.    6?     7?     8?     9?     10?     11?     12?     13? 

4.  In   1   penny  how   many  farthings  ?     2  ?     3  ? 
4?     5?     6?     7?     8?     9?     10?     11?     12?     13? 

5.  In  1  pound  how  many  pennies?     2  ?     3  ?     4  ? 
5?     6?     7?     8?     9?     10?     11?     12?     13? 

6.  In  1  shilling  how  many  pennies  ?     2  ?    3  ?   4  ? 
5?     6?     7?     8?     9?     10?     11?     12?     13?     14? 

7.  In    1    pound   how  many   shillings  ?     2  ?     3  ? 
4?     5?     6?     7?     8?     9?     10?     11?     12? 

Reduction  Ascending. 

8.  In  960  farthings  how  many  pounds  ?     1920? 
2880?     3840?     4800?     5760?     5320? 

9.  In   48   farthings   how   many   shillings  ?     96  ? 
192?     384?     768?     1536?     3072? 


TROY   WEIGHT.  205 

10.  In  4  farthings  how  many  pennies?     8  ?     1C  ? 
32?     64?     128?     256?     512?     1024? 

11.  In  240   pennies  how  many  pounds?     480? 
960?     1920?     3840?     7680?     15360? 

12.  In    12   pennies   how   many   shillings?     24? 
48?     96?     192?     384?     568?     1136? 

LESSON  III. 

296.   TEOY    WEIGHT 

is  used  in  weighing  gold,  silver,  and  diamonds,  etc. 

TABLE. 

24  grains  (grs.)  make  1  pennyweight,  abb.  dwt. 
20  pennyweights  "      1  ounce,  "        oz. 

12  ounces  "      1  pound,  Ib. 

Reduction  Descending. 

1.  In  1  pound  how  many  grains  ?     2?     3?     4? 
5?     6?     7?     8?     9?     10?     11?     12? 

2.  In  1   ounce  how  many  grains  ?     2  ?     3  ?     4  ? 
5?     6?     7?     8?     9?     10?     11?     12? 

3.  In  1  pennyweight  how  many  grains  ?     2  ?     3  ? 
4?     5?     6?     7?     8?     9?     10?     11?     12? 

4.  In  1   pound  how  many  pennyweights  ?     2  ? 
3?     4?     5?     6?     7?     8?     9?     10?     11?     12? 

5.  In   1    ounce   how   many  pennyweights  ?     2  ? 
3?     4?     5?     6?     7?     8?     9?     10?     11?     12? 

6.  In  1  pound  how  many  ounces  ?     2  ?     3  ?    4  ? 
5?     6?     7?     8?     9?     10?     11?     12? 

23 


266 


INTERMEDIATE    ARITHMETIC. 


Reduction  Ascending. 

7.  In  5760  grains  how  many  pounds?     11520? 
17280?     23040?     28800?     74880? 

8.  In    480    grains    how   many   ounces  ?      960  ? 
1440?     1.920?     11040?     12960? 

9.  In  24  grains  how  many  pennyweights  ?    48  ? 
96?     192?     384?     768?     1536? 

10.  In   240  pennyweights   how   many  pounds  ? 
480?     960?     1920?     3840?     7680?     15360? 

11.  In    20    pennyweights    how   many   ounces? 
40?    80?     160?    320?    640?     1280?    2560?    5120? 

12.  In  12  ounces  how  many  pounds  ?     24  ?     48  ? 
96?     192?     384?     768?     1536?     3072?     6144? 

LESSON  IY. 

297".  APOTHECARIES'  WEIGHT 
is  used  in  mixing  medicines. 

TABLE. 


20  grains  (gr.)  make  1  scruple, 
3  scruples  "      1  dram, 

8  drams  "      1  ounce, 

12  ounces  "       1  pound, 


sc.  or  3 
dr.  or  3 
oz.  or  § 
Ib.  or  it). 


5? 


Reduction  Descending. 

1.  In  4  scruples  how  many  grains  ?     6  ?     12  ? 

2.  In  1  pound  how  many  grains  ?     2  ?     3  ?     4  ? 

3.  In    1  ounce  how  many  grains  ?     2  ?     3  ?     4  ? 


APOTHECARIES'  WEIGHT.  267 

4.  In  one  drain  how  many  grains?     2  ?     3  ?     4  ? 
5? 

Reduction  Ascending. 

5.  In  5760  grains  how  many  pounds?     11520? 
23040?     46080?     92160?     184320? 

6.  In    480    grains    Low    many    ounces  ?      960  ? 
1920?     3840?     7680?     15360?     30720? 

7.  In  60  grains  how  many  drams  ?     120  ?     240  ? 
480?     960?     1920?     3840?     7680? 


8.  In   20  grains  how  many  scruples  ?    40? 
3?     320?     640?     1280?     2560? 


80? 
160? 


LESSON  V. 

298.    AVOIEDUPOIS   WEIGHT 

is  used  in  weighing   every  kind  of  goods   having 
weight,  such  as  groceries,  grain,  iron,  copper,  etc. 

TABLE. 

16  drams     (clr.)     make  1  ounce,  oz. 

16  ounces  "     1  pound,  Ib. 

25  pounds  "     1  quarter,  qr. 

4  quarters  "     1  hundred  weight,  cwt. 

20  hundred  weight  "     1  ton,  T. 

EXEECISES. 


euction      escening. 

1.  In  1  ton  how  many  drams?     2  ?     3  ?     4  ? 

2.  In  1  cwt.  how  many  drams  ?     2  ?     3  ?     4  ? 

3.  In  1  quarter  how  many  drams  ?     2  ?    3  ?    4  ? 


268  INTERMEDIATE   ARITHMETIC. 

4.  In  1  pound  tow  many  drams  ?     2  ?     3  ?     4  ? 

5.  In  1  ounce  how  many  drams  ?     2  ?     3  ?     4  ? 

6.  In  1  ton  how  many  ounces  ?    2?    3?    4?    5? 

7.  In  1  cwt.  how  many  ounces  ?    2?     3?   4?   5? 

8.  In  1  qr.  how  many  ounces ?    2?    3?   4?    5? 

9.  In  1  pound  how  many  ounces  ?2?3?    4?5? 

10.  In  1  ton  how  many  pounds  ?    2?   3?   4?   5? 

11.  In  1  cwt.  how  many  pounds?   2?  3?  4?   5? 

12.  In  1  quarter  how  many  pounds  ?      2  ?      3  ? 
4?     5? 

13.  In  1  ton  how  many  quarters?   2?   3?   4?   5? 

14.  In  1  cwt.  how  many  quarters  ?2?3?4?5? 

Reduction  Ascending. 

15.  In  512000  drams  how  many  tons  ?     1024000  ? 
16."  In  25600  drams  how  many  cwt.  ?     51200  ? 

17.  In  6400  drams  how  many  quarters?     12800  ? 

18.  In  256  drams  how  many  pounds  ?     2048  ? 

19.  In  320  drams  how  many  ounces'?     2880  ? 

20.  In  32000  oz.  how  many  tons  ?     288000  ? 

21.  In  1600  oz.  how  many  cwt  ?     11200  ? 

22.  In  400  oz.  how  many  qr.  ?     1600  ?     3200  ? 

23.  In  16  oz.  how  many  pounds?     160?     1280? 

24.  In  2000  Ibs.  how  many  tons  ?  18000  ?  36000? 

25.  In  100  Ibs.  how  many  cwt.  ?     1800  ?     3600  ? 

26.  In  25  Ibs.  how  many  qr.  ?     125  ?     625  ? 

27.  In  80  qr.  how  many  tons  ?     640  ?     1280  ? 

28.  In  4  qr.  how  many  cwt.  ?     16  ?     32  ?     64  ? 

29.  In  20  cwt.  how  many  tons  ?    40?    80?    160  ? 

30.  In  4602000  how  many  tons  ? 


CLOTH   MEASURE.  269 

LESSON  VI. 

299.    CLOTH   MEASURE 

is  used  in  measuring  cloth  and  all  other  articles  sold 
by  the  yard. 

TABLE. 

2j-  inches  (in.)  make  1  nail.  na. 

4    nails                "      1  quarter  of  a  yard.  qr. 

4  quarters          "      1  yard.  yd. 

5  quarters          "      1  Ell  English.  E.  E. 
3    quarters         "      1  Ell  Flemish.  E.  F. 

Reduction  Descending. 

1.  In  1  yard  how  many  inches  ?   2  ?   3?   4?   5? 

2.  In  1  quarter  how  many  inches  ?2?3?4?5? 

3.  In  1  nail  how  many  inches  ?  4?  8?  12?  16? 

Reduction  Ascending. 

4.  In  36  inches  how  many  yards  ?     72  ?     144  ? 
288  ? 

5.  In  9  inches  how  many  quarters?  18  ?  36  ?  72  ? 

6.  In  2£  in.  how  many  nails  ?     8?     12?     16? 

7.  In  16  nails  how  many  yards  ?    32  ?   64  ?    128  ? 

8.  In   4   nails   how   many   quarters?      8?     12? 
24?     48? 

9.  In  4   quarters  how  many  yards?     8?     12? 
36?     72? 

23* 


270 


INTERMEDIATE   ARITHMETIC. 


LESSON  YII. 

30O.    LONG  MEASURE 

is  used  in  measuring  distances. 

TABLE. 

12    inches  (in.)   make  1  foot. 
3    feet  "     1  yard. 

5 1  yards 


yd. 


40    rods 
8    furlongs 


"  1  rod.  r. 

"  1  furlong.  fur. 

"  1  mile.  m. 

3    miles  ct  1  league.  lea. 

69|-  (nearly)  miles  "  1  degree.  deg.  or  °. 

360    degrees  "  1  circumference  of  the  earth. 

circ. 

EXERCISES. 

Reduction  Descending. 

1.  In   1   mile   how   many   feet?      2?     4?     6? 
12?     14? 

2.  In   1   fur.   how  many  feet?      4?     8?     12? 
16?     32? 

3.  In   1  rod  how  many  feet?     8?    12?    16? 
32?     64? 

4.  In  1  yard  how  many  feet  ?9?8?7?6?5? 

5.  In  1  mile  how  many  inches  ?     2  ?     3  ?     4  ? 
5?     6? 

6.  In    1   fur.  how   many  inches?     3?     6?     9? 
12?     15? 

7.  In  1  rod  how  many  inches  ?  6  ?   12  ?   24  ?  48  ? 

8.  In  1  yard  how  many  inches  ?    2  ?    4  ?     8  ? 
16?     32? 


LONG   MEASURE.  271 

9.  In  1  foot  ho  \vmany  inches?  8?  16?  32?  64  ? 

10.  In   1  mile  how  many  yards  ?     3?     6  ?     12  ? 
24?     48? 

11.  In    1    fur.   how  many  yards?     2?     8?     32? 
128  ? 

12.  In  1  rod  how  many  yards  ?     6  ?     36  ?     216  ? 

13.  In  1  mile  how  many  rods  ?  8  ?  16  ?  32  ?    64  ? 

14.  In   1    fur.   how   many   rods?      2?     4?     5? 
6?     7? 

15.  In  1  circle  how  many  inches  ?     2  ?     3  ?     4  ? 
5?     6? 

Reduction  Ascending. 

16.  In  63360  in.  how  many  miles  ?  126720  ? 

17.  In  7920  inches  how  many  fur.  ?  23760  ? 

18.  In  198  inches  how  many  rods?  792  ? 

19.  In  36  inches  how  many  yards  ?  180  ? 

20.  In   5280   feet   how   many   miles?      10560? 
21120? 

21.  In  660  feet  how  many  furlongs?  1980?  5940? 

22.  In  16i  feet  how  many  rods  ?     66  ?     528  ? 

23.  In  3  feet  how  many  yards  ?     27  ?    54  ?    108  ? 

24.  In  1760  yards  how  many  miles?  3520?   7040? 

25.  In   220   yards   how   many    furlongs?     440? 
880?     1760? 

26.  In  5£  yards  how  many  rods  ?    22  ?    44  ?    88  ? 

27.  In  920  rods  how  many  miles  ?     640  ?     1280  ? 
2560? 

28.  In  40  rods  how  many  furlongs?     80?     160? 
320?     640? 

29.  In  8  furlongs  how  many  miles?     16?    32? 
64?     128? 


272 


INTERMEDIATE   ARITHMETIC. 


1  link. 

I 

1  pole. 
1  chain. 

p. 

cha. 

1  chain. 

cha. 

1  chain. 

cha. 

1  furlong. 
1  mile. 

fur. 
mi. 

1  mile. 

mi. 

LESSOR    VIII. 

3O1.  SURVEYOR'S  MEASURE 
is  used  in  measuring  land. 

TABLE. 

8  inches  (nearly)  (7. 92)  make  1  link. 
25  links 
100  links 

4  poles 
66  feet 
10  chains 
80  chains 

8  furlongs 

EXERCISES. 
Reduction  Descending. 

1.  In  1  mile  how  many  inches ?    2?  3?  4?    5? 

2.  In  1  furlong  how  many  links  ?  2?  4?  8?  16? 

3.  In  1  chain  how  many  links ?  2?  9?  13?  26? 

4.  In  1  furlong  how  many  poles  ?  4?  8?  9?  12? 

Reduction  Ascending. 

5.  In  8000  links  how  many  miles  ?     160000  ? 

6.  In  80  chains  how  many  furlongs  ?  640?  1280? 

7.  In  80  chains  how  many  miles  ?    640  ?    2560  ? 

8.  In  320  poles  how  many  miles?     640  ?    1280? 


SQUARE  MEASURE.  273 

LESSON  IX. 

3O2.    SQUARE  MEASURE. 

Square  measure  is  the  square  of  Long  Measure, 
and  is  used  in  measuring  any  thing  which  has  length 
and  breadth. 

TABLE. 

144    inches  (in.)  make  1  square  ft.       sq.ft. 

9    feet  "      1       "      yd.    sq.yd. 

301-  yards  "      1       u       rod.     sq.  r. 

40    rods  "      1       <£       rood.       It. 

4^  roods  "      1       "      acre.        A. 

640    acres  "      1  square  mile.   sq.m. 

EXERCISES. 
Reduction  Dssccnding. 

1.  In  1  square  mile  how  many  inches  ?     2  ?     3  ? 

2.  In  1  acre  how  many  inches  ?     2  ?     4  ?     6  ? 

3.  In  1  rood  how  many  inches  ?     2  ?     3  ?     4  ? 
5?     6? 

4.  In  1  rod  how  many  inches  ?     3  ?    5  ?     6  ? 
7?     8? 

5.  In  1  yard  how  many  inches  ?     5  ?     6  ?     7  ? 
8?     9? 

6.  In   1   foot  how   many  inches?      10?      11? 
12?     13? 

7.  In  1  mile  how  many  feet  ?     2?     3?    4?    5? 
6?     7? 

8.  In  1  acre  how  many  feet  ?  3?  4?5?  7?    9? 

9.  In   1   rood  how  many  feet  ?     4  ?     6  ?     8  ? 
10?     11? 


274  INTERMEDIATE   ARITHMETIC. 

10.  In   1    rod   how   many   feet?     3?     6?     12? 
24?     5? 

11.  In  1  yard  how  many  feet?     2?     3?     4?     5? 
6?     7? 

12.  In   1    acre  how  many  yards?     3?     4?     5? 
6?     7?     8? 

13.  In  1   mile  how  many  yards?     4?     5?     6? 
7?     8?     9? 

14.  In  1   rood  how  many  yards?     2?     3?     4? 
5?     6?     7? 

15.  In   1   rod  how   many   yards?     3?     5?     7? 
9?     11?     13? 

16.  In   1  mile   how   many  rods?     3?  '4?     5? 
6?     7?     8? 

17.  In   1    acre  how  many  rods?     4?     5?    6? 
7?     8?     9? 

18.  In    1   rood  how  many  rods?     3?     8?     9? 
18?     30?     32? 

19.  In  1  mile  how  many  roods?     2?     3?    4? 
5?     6?     7?     8?     9? 

20.  In   1   acre  how  many  roods?     2?    3?    4? 
5?     6?     7?     8?     9? 

Reduction  Ascending. 

21.  In  8028979200  inches  how  many  miles  ? 

22.  In  627264000  inches  how  many  acres  ? 

23.  In  15681600  inches  how  many  roods  ? 

24.  In  39204000  inches  how  many  poles? 

25.  In  1296000  inches  how  many  yards  ? 

26.  In  278784000  feet  how  many  miles  ? 

27.  In  43560000  feet  how  many  acres? 


SQUARE  MEASURE.  275 

28.  In  1089000  feet  how  marly  roods  ? 

29.  In  272£  feet  how  many  poles  ? 

30.  In  999  feet  how  many  yards? 

31.  In  3097600  yards  how  many  miles  ? 

32.  In  48404840  yards  how  many  acres  ? 

33.  In  12101210  yards  how  many  roods  ? 

34.  In  1024000  poles  how  many  acres  ? 

35.  In  160160160  poles  how  many  acres? 

36.  In  40404040  poles  how  many  roods  ? 

37.  In  25602560  roods  how  many  miles  ? 

38.  In  40404040  roods  how  many  acres? 

39.  In  640640640  acres  how  many  miles? 

40.  In  40144896004014489600  inches  how  many 
miles. 

LESSOR  X. 

3O3.  A  square  is  a  figure  having  4  equal  sides 
and  4  right  angles. 

2  square  feet  and  2  feet  square  are  not  the  same. 
2  feet  square  is  a  body  2  feet  long  and  2  feet  broad, 
which  is  equal  to  4  square  feet.  2  square  feet  is  £ 
of  4  square  feet. 

Pupils  always  make  mistakes  in       2  feet  square, 
not  being  able  to  make  any  differ- 
ence between  such  expressions  as 
the  following :  6  square  feet  and  6 
feet  square,  2  yards  square  and  2 
square  yards.    Pupils  would  require 
drilling  on  this  subject,  and  for  that  reason  I  have 
here  inserted  a  few  questions. 


276  INTERMEDIATE   ARITHMETIC. 

Explain  the  meaning  of  the  following  terms,  and 
show  their  difference. 

6  ft.  square  and  6  square  feet. 
2  ft.  square  and  2  square  feet. 
5  yds.  square  and  5  square  yards. 
2  miles  square  and  2  square  miles. 
4  miles  square  and  16  square  miles. 

LESSON  XL 

'IO4.    CUBIC   OK   SOLID   MEASURE 

is  used  in  measuring  bodies  having  length,  breadth, 
and  thickness,  such  as  stone,  timber. 

TABLE. 

1728  cubic  inches  (cu.  in.)  make  1  cubic  foot,    cu.ft. 

27  cubic  feet  "     1  cubic  yard,  cu.  yd. 

40  cubic  feet  "     1  Ton,  T. 

16  cubic  feet  "      1  cord  foot,       c.  ft. 

8  cord  feet  or  "     1  cord  of  wood,     C. 

128  cubic  feet  "     1  cord  of  wood,     C. 

Reduction  Descending  and  Ascending. 

1.  In  1  cord  how  many  inches  ?   2?   3?   4?   5  ? 

2.  In  1  cord  how  many  feet  ?     2?     3  ?     4?     5  ? 

3.  In  1  ton  how  many  inches  ?2?3?     4?     5? 

4.  In  1  ton  how  many  feet  ?     2?     3?     4?     5? 
6? 

5.  In  221184  inches  how  many  cords  ?   442368  ? 
884736? 

6.  In  69120  inches  how  many  tons  ?     128240  ? 
256480? 


CUBIC    OR   SOLID   MEASURE. 


277 


7.  In  4665G  inches  how  many  yards?     93312  ? 
186624  ? 

8.  In  172  8  inches  how  many  feet?   3456?   6912? 

9.  In  128  feet  how  many  cords  ?     256?     512? 
1024? 

10.  In  40  feet  how  many  tons  ?   80?    160?   320? 

11.  In  2 7  feet  how  many  yards?    54?    108?   216? 


LESSON  XII. 

3O«>.  A  cube  is  a  solid   body  having   length, 
breadth,  and  thickness.     A  cube  has  6  sides. 


To  find  the  solidity  of  a  cube,  multiply  the 
length,  breadth,  and  height  together,  or,  in  other 
words,  multiply  the  length  of  one  side  by  itself  two 
times. 

EXAMPLES. 

1.  How  many  inches  in  a  square  block  of  wood, 
1  foot  in  length,  breadth,  and  height. 

Ans.  1728  cu.  in. 
OPEBATION. 

12X12X12  =  1728. 
24 


278  INTERMEDIATE  ARITHMETIC. 

2.  What  is  the  solidity  of  a  piece  of  timber  20 
feet  long,  3  feet  broad,  and  4  feet  high  ? 

20  X  3  X  4  ~  240  cubic  feet. 

3.  What  is  the  solid  contents  of  a  cube  of  ice,  2 
feet  high  ? 

2X2X2=8  cubic  feet. 

4.  What  is  the  solidity  of  a  square  log,  40  feet 
long,  1  foot  high,  and  2  feet  broad  ? 

40  x  1  X  2  —  SO  cubic  feet. 

5.  How  many  cubic  feet  of  earth  was  removed  to 
make  a  square  well,  40  feet  deep,  and  5  feet  square 
at  the  top  ? 

40  X  5  x  5  =  1000  cubic  feet. 


LESSON  XIII. 

SO 6.   WINE   MEASURE 

is  used  in  measuring  all  kinds  of  liquids,  except 
beer,  milk,  ale,  or  porter. 

TABLE. 


4  gills  (gi.)  make 

1  pint,         abb. 

pt. 

2  pints             " 

1  quart, 

gt. 

4  quarts 

1  gallon, 

gal. 

63  gallons 

1  hogshead, 

hdd. 

2  hogsheads    " 

1  pipe, 

pi. 

2  pipes 

1  tun, 

tun. 

Reduction  Descending. 

1.  In  1  tun  how  many  gills  ?     2?     4?     6?     8? 

2.  In  1  tun  how  many  pints  ?    2  ?    3  ?    4  ?    5  ? 


WINE  MEASURE.  279 

3.  In  2  tuns  how  many  quarts ?    6?   7?   8?   9? 

4.  In  3  tuns  how  many  gallons  ?     10?    11?    12? 
13? 

5.  In  4  tuns  how  many  pipes  ?     14?     15?     10? 
17? 

6.  In   2   pipes  how  many  gills  ?    3  ?    4  ?    5  ? 
6?     7? 

7.  In  2  pipes  how  many  pints  ?    4  ?    5  ?    G  ? 
7?     8? 

8.  In  1  pipe  how  many  quarts  ?     6  ?     7  ?    8  ? 
9?     10? 

9.  In  2  pipes  how  many  gallons  ?    24  ?    2G  ?  30  ? 

10.  In  2  pipes  how  many  hogsheads  ?     21  ?    22  ? 
23? 

11.  In  1  hhd.  how  many  gills  ?    2?    3?    4?    5? 
6? 

12.  In  2  hhds.  how  many  pints  ?    3  ?    4  ?    5  ? 
6?     7? 

13.  In  3  hhds.  how  many  quarts?     4?    5?    6? 
7?    8? 

14.  In  4  hhds.  how  many  gallons  ?    5  ?    6  ?     7  ? 
8?     9? 

15.  In  2  galls,  how  many  gills?    6?    7?    8? 
9?     10? 

16.  In   3   galls,  how  many  pints  ?     7  ?     8  ?    9  ? 
10?     11? 

17.  In  4  galls,  how  many  quarts  ?   3?  4?   5?  6? 

18.  In  1  quart  how  many  gills  ?     2?    3?    4?.  5? 

19.  In  2  quarts  how  many  pints?    2?  3?   4?  6? 

20.  In  2  pints  how  many  gills  ?     2?     3?    4?    7? 


280  INTERMEDIATE    ARITHMETIC. 

Reduction  Ascending. 

1.  In    16128   gills   how   many   tuns?      32256? 
64512? 

2.  In  4032  gills  how  many  pipes  ?  8064?  16128? 

3.  In    4032    gills    how    many   hhds.  ?      8064? 
16128? 

4.  In  320  gills  how  many  galls.  ?  1820  ?  10920  ? 

5.  In  72  gills  how  many  quarts?   720  ?    14400  ? 

6.  In   4032   pints    how   many   tuns  ?      12096  ? 
36288  ? 

7.  In   2016   pints    how  many   pipes?     20160? 
141120? 

8.  In    1008    pints    how   many   hhds.  ?      2016  ? 
181440? 

9.  In  64  pints  how  many  galls.  ?     128  ?     256  ? 
512? 

10.  In    4   pints   how   many   quarts?     16?     32? 
64?     128? 

11.  In   2016   quarts    how  many   tuns'?      8064? 
32256?     129024? 

12.  In   1008   quarts  how  many  pipes  ?     6048  ? 
36288? 

13.  In   1512   quarts   how   many   hhds.?     3024? 
6048?     12096? 

14.  In  44  quarts  how  many  galls.  ?     88  ?     792  ? 
7128? 

15.  In    2520    galls,   how  many  tuns?      5040? 
10080? 

16.  In   1260   galls,   how  many  pipes  ?      2520  ? 
5040?     10080? 


BEER   MEASURE.  281 

17.  In   630    galls,   how  many   hhds.?       1260? 
2520?     5040? 

18.  In  444  hhds.  how  many  tuns  ?     888  ?     7004  ? 
56032  ? 

19.  In  222  hhds.  how  many  pipes  ?    666  ?    1332  ? 
2664? 

20.  In  44   pipes  how  many   tuns?     88?     176? 
352  ? 

21.  In   20160   pints   how  many  hhds.?     40320? 
80640? 

LESSON  XIV. 

3O7.    BEER   MEASURE 

is  used  in  measuring  beer,  ale,  and  porter.     Milk  is 
measured  by  the  beer  measure. 

TABLE. 

2  pints  (pt.)  make  1  quart.  qt. 

4  quarts  "      1  gallon.        gal. 

54  gallons          "      1  hogshead,  hhd. 

Reduction  Ascending. 

1.  In   43200   pints    how   many   hhds.  ?     8640  ? 
17280? 

2.  In  88  pints  how  many  galls.?  176?  352?  704? 

3.  In  880  pints  how  many  galls.  ? 

4.  In  216  quarts  how  many  hhds.  ?    432  ?    864  ? 
1728? 

5.  In   40   quarts   how   many   gallons?      1600? 
32000? 

6.  In  54  galls,  how  many  hhds.  ?    1080?   2160? 

24* 


282 


INTERMEDIATE   AEITHMETIC, 


Reduction  Descending. 

1.  In   1    hhd.   how  many  pints  ?     2  ?     3  ?     4  ? 
5?     6?     7? 

8.  In  1  gallon  how  many  pints  ?     3  ?     4  ?     5  ? 
6?     7?     8?     9? 

9.  In  2  quarts  how  many  pints  ?     4  ?     5  ?     C  ? 
7?     8?     9?     10? 

10.  In  1   hhd.  how  many  quarts?     2?     3?     4? 
5?     C?     7? 

11.  In  1  gallon  how  many  quarts  ?     3  ?    4  ?     5  ? 
6?     7?     8? 

12.  In  4  gallons  how  many  pints  ?     5  ?     6  ?     7  ? 
8?     9?     10? 

NOTE. — Beer  measure  is  becoming  obsolete.     Milk  and  malt 
liquors  are  now  measured  by  the  wine  measure. 


LESSOR  XV. 

3O8.     DRY   MEASURE 


is  used  in  measuring  fruit  and  grain.     Coal  and  salt 
are  also  measured  by  Dry  Measure. 


TABLE. 


2  pints  (pt.)  make  1  quart. 
8  quarts 
4  pecks 
8  bushels 
36  bushels 
4  quarters 


qt. 
pk. 
bu. 


1  peck. 
"  1  bushel. 
"  1  quarter.  qr. 
"  1  chaldron.  ch. 
"  1  load.  L. 


DRY   MEASURE.  283 

Reduction  Descending. 

1.  In    1    qr.   how   many   pints?      2?      3?     4? 
5?     G? 

2.  In    1   bu.   how   many   pints  ?      3  ?      4  ?     5  ? 
G?     7? 

3.  In   1    pk.   how   many   pints  ?      7  ?      8  ?     9  ? 
10?     11? 

4.  In    1    qt.   how   many   pints  ?      4  ?      5  ?     G  ? 
7?     8? 

5.  In   1    qr.  how   many   quarts  ?     2  ?     3  ?     4  ? 
5?     6? 

6.  In   1   bu.   how  many  quarts  ?     7  ?     8  ?     9  ? 
10?     11? 

7.  In   1    peck   how   many   quarts  ?      9  ?      10  ? 
11?     12? 

8.  In  1  qr.  how  many  pecks  ?     2?     3?     4?    5? 

9.  In  1  bu.  how  many  pecks  ?     6  ?    7  ?     8  ?     9  ? 
^10.  In  1  qr.  how  many  bushels  ?    6?8?10?12? 

Reduction  Ascending. 

11.  In  2304  pints  how  many  chaldrons  ?      4608  ? 
9216? 

12.  In    640  pints   how   many   bushels  ?      1280  ? 
2560? 

13.  In  160  pints  how  many  pecks?     320?     640? 

14.  In    11520     quarts    how    many    chaldrons? 
23040  ?     46080  ? 

15.  In   320   quarts    how  many  bushels?     640? 
1280?     2560? 


284 


INTERMEDIATE   ARITHMETIC. 


16.  In  80  quarts  how  many  pecks  ?     160  ?     320  ? 
6400? 

17.  In  2880  pecks  how  many  chaldrons  ?     5760  ? 
11520  ? 

18.  In  48  pecks  how  many  bushels  ?     96  ?     192  ? 

384? 

.    19.  In  720  bushels  how  many  chaldrons  ?     1740  ? 
3480  ?     6960  ? 

20.  In  1440  bushels  how  many  chaldrons  ?  2880  ? 
5760? 


LESSON  XVI. 


3O9.    TIME    TABLE. 


60  seconds  (sec.) 
60  minutes 
24  hours 
7  days 
52  weeks 
100  years 
12  months 


make  1 

minute, 

m. 

"      1 

hour, 

hr. 

"      1 

day, 

d. 

"      1 

week, 

w. 

"      1 

year, 

yr. 

"      1 

century, 

C. 

«      1 

year, 

yr. 

)   "      1 

month, 

mo. 

31O.  The  Tropical  year  is  365  days,  5  hrs.  48 
m.  and  49  sec. 

The  Julian  Year,  called  after  Julius  Ca3sar,  is  365 
days  and  ^  of  a  day. 

The  Gregorian  Year,  named  after  Pope  Gregory 


TIME    TABLE.  285 

XIII.,  and  which  is  the  most  commonly  used  by  all 
of  the  nations,  gives  365  days  for  every  3  consecutive* 
years,  and  300  days  for  every  fourth  year.  As  far 
as  ascertained,  this  is  the  most  correct. 

A  Sidereal  Year  is  the  length  of  time  the  earth 
takes  in  revolving  around  the  sun,  and  is  365  days, 
6  hrs.  9m.  9  sec. 


.  The  12  months  of  a  year  are  named: 
1st.  January.     2d.  February.      3d.  March.     4th. 
April.     5th.  May.     6th.  June.     7th.  July.     8th.  Au- 
gust.    9th.  September.     10th.  October,     llth.  No- 
vember.    12th.  December. 

312.  The  months  have  not  the  same  number  of 
days. 

January  has  31,  February  28,  March  31,  April 
30,  May  31,  June  30,  July  31,  August  31,  Septem- 
ber 30,  October  31,  November  30,  and  December  31. 

313.  Every  leap  year  gives  February  29  days. 

314.  It  would  be  well  to  commit  to  memory 
the  following  verse,  to  assist  in   remembering  the 
number  of  days  in  each  month. 

Thirty  days  hath  September, 
April,  June,  and  November, 
All  the  rest  have  thirty-one, 
Except  February,  which  alone 
Hath  twenty-eight  ;  and  this  in  fine, 
Till  the  leap  year  gives  it  29. 


286 


INTERMEDIATE   ARITHMETIC. 


315.     TABLE 


SHOWING  THE    XT73IBEF.    OP  DAYS    FEOH    ANY   DAY   OP    ONE    MONTH    TO    THE 
SAME  DAY  OF  ANY  OTHEE  MONTH  OF  THE  SAME  YISAU. 


FROM 
ANY  DAY  OF 

TO  THE  SAME  DAY  OF 

Jan. 

Feb 

Mar. 

Apr. 

Mny. 

June 

July. 

Aug.  Sept. 

Oct. 

Nov 

Dee. 

1.  January.  . 
2.  February  . 
3.  March  
4  A'oril. 

365 
334 
306 
275 
245 
214 
184 
153 
122 
92 
61 
31 

31 
365 
337 
306 
276 
245 
215 
184 
153 
123 
92 
62 

59 

28 
365 
334 
304 
273 
243 
212 
181 
151 
120 
90 

90 
59 
31 
365 
335 
304 
274 
243 
212 
182 
151 
121 

120 
89 
61 
30 
365 
334 
304 
273 
242 
212 
181 
151 

151 
120 
92 
61 
31 
365 
335 
304 
273 
243 
212 
182 

181 

150 
122 

91 
61 
30 
365 
334 
303 
273 
242 
212 

212 
181 
153 
122 
92 
61 
31 
365 
334 
304 
273 
243 

243 
212 
184 
153 
123 
92 
62 
31 
365 
335 
304 
274 

273 
242 
214 
183 
153 
122 
92 
61 
30 
365 
334 
304 

304 
273 
245 
214 
184 
153 
123 
92 
61 
31 
365 
335 

334 

303 
275 
244 
214 
188 
153 
122 
91 
61 
30 
365 

5  May  

G.  June  

7  July 

8.  August..  . 
9.  September 
10.  October  .  . 
11.  November 
12.  December. 

316.  To  find  the  number  of  days  from  any  day 
of  one  month  to  any  day  of  any  other  month  of  the 
same  year. 


EXAMPLE. 


1.  How  many  days  from  Feb.  1st  to  Oct.  1st. 
February  is  the  second  month.     Run  along  the 

second  line  of  figures  parallel  to  the  top  of  the  page, 
until  you  come  to  the  month  of  October.  The  num- 
ber 242,  in  the  vertical  October  column,  represents 
the  number  of  days  from  February  to  October. 

2.  How  many  days  from  Feb.  1st* to  Oct.  9th. 
First  find  how  many  days  from  Feb.  1st  to  Oct. 

1st.  It  is  242  days.  But  that  is  not  enough  of 
days,  because  we  find  the  number  of  days  to  Oct.  1st. 


TIME    TABLE.  287 

To  Oct.  9th  would  be  8  clays  more.  Add  8  days  to 
242  days,  and  the  sum  is  250  for  the  number  of  days 
from  Feb.  1st  to  Oct.  9th. 

3.  How  many  days  from  Feb.  9th  to  Oct.  1st. 

First  find  from  Feb.  9th  to  Oct.  9th,  which  is  242 
days.  But  that  is  too  many  days,  because  we  found 
the  number  of  days  9th  of  October.  The  number 
of  days  too  many  would  bo  the  difference  between  9 
and  1,  that  is,  the  number  of  days  between  the  1st 
and  9th  of  October.  The  difference  is  8  days,  which 
must  be  subtracted  from  242.  242 — 8  =  234  days, 
the  number  of  days  from  Feb.  9th  to  Oct  1st,  1869. 

EXAMPLES. 

1.  How  many  days  from  Jan.  1st  to  Dec.  10th. 

Ans. 

2.  How  many  days  from  Feb.  14th  to  Dec.  23d. 

Ans. 

3.  How  many  days  from  March   24th  to  Sept. 
2Gth.  Ans. 

4.  How  many   days  from  April  2Gth  to  Sept. 
25th.  Ans. 

5.  How  many  days  from  Jan.  10th  to  May  2d  ? 

Ans. 

6.  How   many  days   from   Feb.   20th   to   Sept. 
26th  ?  Ans. 

Reduction  Descending. 

1.  Iii  1  year  how  many  minutes  ?     2  ?     3  ?     4  ? 
5? 

2.  In    1    month   how   many   minutes  ?     6  ?     7  ? 
8?     9? 


288 


INTERMEDIATE    ARITHMETIC. 


3.  In    1    day   how   many   minutes?     10? 
12?     13? 

4.  In  1  hour  how  many  minutes?      13? 
15?     1G? 

5.  In  1  year  how  many  hours  ?     2  ?    3  ?    6  ? 

6.  In  1  month  how  many  hours  ?     3  ?     8  ? 
7? 

7.  In   1    week  how  many  hours?     6  ?     8  ? 
11? 

8.  In   1   day  how  many  hours  ?     2  ?     8  ? 
14? 

9.  In  1  year  how  many  days  ?    3?    4?    14? 

10.  In    one    month   how   many   days?      6? 
8?     9? 

11.  In    one    year   how  many   months?     3? 
9?     10? 

12.  In  1  Century  how  many  minutes  ? 

Reduction  Ascending. 

13.  In  315576000  seconds  how  many  years? 

14.  In  6048000  seconds  how  many  weeks  ? 

15.  In  864000  seconds  how  many  days  ? 

16.  In  36000  seconds  how  many  hours? 

17.  In  52596000  minutes  how  many  years  ? 

18.  In  100800  minutes  how  many  months  ? 

19.  In  144000  minutes  how  many  days  ? 

20.  In  876600  hours  how  many  years  ? 

21.  In  168000  hours  how  manv  months? 


11? 
14? 

7? 
9? 

10? 

12? 

5? 


CIRCULAR    MEASURE. 


289 


22.  In  24000  hours  how  many  days  ? 

23.  In  365|  days  how  many  years  ?     1461  ? 

24.  In  700  days  how  many  weeks  ?     1400  ? 

25.  In  52  weeks  how  many  years  ?     104  ? 


LESSON  XVII. 

317.   CIRCULAR   MEASURE 

is  used  in  measuring  circles  and  angles.  It  is  also 
used  in  reckoning  Latitude  and  Longitude,  and  the 
revolutions  of  the  heavenly  bodies  around  the  earth. 

TABLE. 

60  seconds  (")  make  1  minute, 

60  minutes  (')        "  1  degree,  ° 

30  degrees             "  1  sign,  S. 

12  signs                  "  1  circle,  C. 

360  degrees             "  1  circle,  C. 


318.  An  angle  is  the  opening  be- 
tween two  lines  which  meet,  thus  : 


A  right  angle  is  formed  by  one  line 
falling  perpendicular  to  another.    Thus : 


31O.  An  acute  angle  is  a  smaller 
angle  than  a  right  angle.     Thus, 


290 


INTERMEDIATE   ARITHMETIC. 


An  obtuse  angle  is  a  larger  angle 
than  a  right  angle.     Thus, 


The  point  A,  where  the  line  meets,  is  called  the 
Vertex. 

3*2O.  A  Circle  is  a  plane  bounded  by  a  curved 
line,  every  point  of  which  is  equally  distant  from  a 
point  within,  called  the  center. 

321.  The    Circumference    is    the   curved    line 
which  bounds  the  plane. 

322.  An  Arc  is  any  part  of  the  circumference. 

323.  In    a  circle  there  are  four  right  angles, 
therefore  a  right  angle  is  equal  to  one  quarter  of  a 
circle,  and  since  a  circle  is  divided  into  360  degrees, 
a  right  angle  is  equal  to  J  of  360  degrees,  or  90 
degrees. 

324.  A  quadrant  is  |-  of  a  circle. 

325.  The  Diameter  of  a  circle  is  a  straight  line 
which  divides  the  circle  into  two  equal  parts. 

326.  Two   diameters    crossing    each   other    at 
right  angles,  divide  the  circle  into  four  equal  parts. 

327.  The    radius    of   a 
circle   is  i  of  the  diameter, 
that  is,  that  part  of  the  di- 
ameter  between   the   center 
and   the   circumference.      In 
the  circle  to  the  right  there 
are   4   radii.     (Radii    is   the 
plural  of  Radius.) 

In  any  circle  there  are  twice  as  many  radii  as 
there  are  diameters. 


CIRCULAR   MEASURE.  291 

328.  An  Arc  is  the  measurement  of  an  angle. 
Thus  an  arc  between  two  lines,  one  perpendicular  to 
the  other,  is  equal  to  90  degrees.  The  angle  formed 
by  those  two  lines  is  therefore  90  degrees. 

320.  The  sun  turns  on  its  axis  once  in  24  hours, 
from  west  to  east,  that  is,  the  sun  passes  over  15°  of 
longitude  in  every  hour,  or  1°  in  every  4  minutes  of 
time,  and  1'  in  every  4  seconds  of  time.  Hence, 
when  it  is  noon  or  12  o'clock  at  New  Orleans,  it  is 
evening  in  all  cities  east,  and  morning  in  all  cities 
Avest  of  that  place. 

Reduction  Descending. 

1.  In  1  circle  how  many  seconds?     2  ?     3  ?    4  ? 

2.  In  1  sign  how  many  seconds  ?     2  ?     3  ?     4  ? 

3.  In  1  degree  how  many  seconds  ?     2  ?  3  ?   4  ? 

4.  In  1  minute  how  many  seconds  ?    2  ?   3  ?    4  ? 

5.  Iu  1  circle  how  many  minutes  ?     2  ?     3  ?    4  ? 

6.  In  1  sign  how  many  minutes  ?     2  ?     3  ?     4  ? 

7.  In  1  degree  how  many  minutes  ?    2  ?   3  ?   4  ? 

8.  In  1  circle  how  many  degrees  ?     2  ?     3  ?    4  ? 

9.  In  1  sign  how  many  degrees  ?     2  ?     3  ?     4  ? 

10.  In  1  circle  how  many  signs  ?     2  ?     3  ?    4  ? 

Redaction  Ascending. 

11.  In  1296000  sec.  how  many  circles  ?  5184000  ? 

12.  In  108000  sec.  how  many  signs  ?     648000? 

13.  In  3600  sec.  how  many  degrees  ?     324000  ? 

14.  In  21600'  how  many  circles  ?     43200  ? 

15.  In  1800'  how  many  signs  ?     7200  ? 

16.  In  600'  how  many  degrees?     3000  ? 


292  INTERMEDIATE   ARITHMETIC. 

17.  In  3600°  how  many  circles?     18000? 

18.  In  300°  how  many  signs  ?     15000  ? 

19.  In  12  signs  how  many  circles  ?     4800? 

20.  In  120  signs  how  many  circles  ?     60000  ? 


LESSON  XVIII. 

.    DUODECIMALS 

are  compound  numbers  whose  denominations  increas 
or  decrease  in  a  tAvelvefold  ratio. 

It  is  used  in  measuring  surfaces  and  solids  such 
as  paving,  plastering,  etc. 

The   denominations    are    foot    (ft.),   prime    ('), 
second  (  "  ),  third  (  "' ). 

TABLE. 

12'"  (thirds)  make  1  second. 

12"  "      1  prime. 

12'  "1  foot.  ft. 

EXEKCISES. 

1.  In  17280'"  how  many  feet? 

2.  In  1440"  how  many  feet  ? 

3.  In  2  feet  how  many  seconds. 

4.  In  20  feet  how  many  thirds  ? 

5.  In  20  primes  how  many  thirds  ? 

6.  In  12  seconds  how  many  thirds  ? 

7.  In  25  feet  how  many  primes  ? 

8.  How  many  feet  in  1728  primes  ? 

9.  How  many  primes  in  144  seconds  ? 
10.  How  many  seconds  in  25  feet  ? 


REDUCTION  DESCENDING.  293 

LESSON  XIX. 

331.  MISCELLANEOUS   TABLE. 

12  units  make  1  dozen. 

12  doz.  or  144  units  "  1  gross. 

12  gross  or  1728  units  "  1  great  gross. 

20  units  "  1  score. 

56  Ibs.  corn  shelled  "  1  bushel. 

196  Ibs.  flour  "  1  barrel. 

24  sheets  paper  "  1  quire. 

20  quires  "  1  ream. 

2  reams  "  1  bundle. 

5  bundles  or  10  reams  "  1  bale. 

A  sheet  of  paper  folded  in  2  leaves  makes  a  folio. 
"  "  "        4       "          "         4to. 

"  "  "         8       "  "          8vo. 

<c  «  «       12       «  «          12mo. 

"  "  "18       "  "          18mo. 

LESSON  XX. 

332.  REDUCTION  DESCENDING 

when  there  are  several  units  of  different  denomina- 
tions. 

EXAMPLE. 

Reduce  2£  3s.  4d.  2  far.  to  farthings. 

Begin  at  the  highest  denomination,  £,  and  reduce 
it  to  shillings :  2  x  20=40  shillings,  to  which  adding 
3  gives  43  shillings.  Reduce  the  43  shillings  to 

25* 


294 


INTERMEDIATE   ARITHMETIC. 


pence  by  multiplying  by  12  :  43  x  12  =  516.  To 
which  add  4d.,  making  520d.  Reducing  520  pence 
to  farthings  gives  520  x  4=2080  farthings,  to  which 
adding  2  farthings  gives  2082  farthings  for  the 
answer. 

EULE. 

I.  Arrange  the  several  denominations  in  a  hori- 
zontal line. 

II.  Multiply  the  highest  term  of  the  denominate 
number  by  the  number  of  the  next  lower  denomina- 
tion required  to  make  one  of  that  higher  one,  and  to 
the  product  add  the  term  in  the  given  quantity,  which 
is  of  the  same  denomination  as  the  product,  and  con- 
tinue the  work  as  above  till  the  lowest  given  units 
have  l)een  added. 

1.  Reduce  27  Ibs.  15  dwt.  to  dwt. 

Ans.  6495  dwt. 

2.  Reduce  2£  15s.  9d.  to  farthings. 

Ans.  2676  far. 

3.  Reduce  20£  5s.  to  farthings.     Ans.  19440  far. 

4.  Reduce  20£  15s.  to  farthings.  Ans.  19220  far. 

5.  Reduce  17s.  5  far.  to  farthings.     Ans.  821  far. 

6.  Reduce  1  Ib.  2  02.  3  dwt.  2  gr.  to  grains. 

Ans.  6794  grs, 

7.  Reduce  1  Ib.  3  oz.  5  dwt.  to  grains. 

Ans.  7320  grs. 

8.  Reduce  3  oz.  15  dwt,  to  grains. 

Ans.  1560  grs. 

9.  Reduce  7  Ton  5  oz.  .to  oz.        Ans.  224005  oz. 
10.  Reduce  17  Ton  15  oz.  to  oz.    Ans.  544015  oz. 


KEDUCTION   DESCENDING.  295 

11.  Reduce  13  Ton  7  cwt.  3  qr.  to  oz. 

Ans.  428400  oz. 

12.  Reduce  13  Ton  17  cwt.  2  qrs.  to  quarters. 

Am.  lilO  quarters. 

13.  Reduce  1  qr.  10  oz.  3  dr.  to  drams. 

Am.  6563  drs. 

14.  Reduce  5  rods  3  yds.  2  ft.  G  in.  to  in. 

Ana.  1128  in. 

15.  Reduce  4  rods  2  yds.  1  ft.  2  in.  to  in. 

Am.  878  in. 

16.  Reduce  22  fur.  6  rods  3  yds.  2  ft.  to  ft. 

Ans.  14630  ft. 

17.  Reduce  4  mi.  6  fur.  20  r.  16  in.  to  in. 

Ans.  101656  in. 

18.  Reduce  2  yds.  1  qr.  2  na.  to  inches. 

Ans.  85 £  in. 

19.  Reduce  4  ft.  6  in.  to  inches.  Ans.  54  in. 

20.  Reduce  1  Tun  1  gall.  1  qt.  to  quarts. 

Ans.  1013  quarts. 

21.  Reduce  2  Tuns  1  gall.  3  qrts.  to  gills. 

Ans.  16184  gills. 

22.  Reduce  3  Tuns  2  galls,  to  gills. 

Ans.  24256  gills. 

23.  Reduce  40  bbls.  18  galls,  to  pints. 

Ans.  10224  pints. 

24.  Reduce  15  bu.  2  pks.  to  quarts. 

Ans.  506  quarts. 

25.  Reduce  14  A.  1  R.  to  yards.   Ans.  68970  yds. 


296  INTERMEDIATE   ARITHMETIC. 

26.  Reduce  14  A.  2  R.  to  yards.        Am.  58  rods. 

27.  Reduce  IV  A,  3  R.  2  r.  to  inches. 

Ans.  111417768  in. 

28.  Reduce  14  A.  1  R.  to  feet.        Ans.  620730  ft. 

29.  Reduce  1  sq.  m.  2  in.  to  inches. 

Ans.  4014489602. 

30.  Reduce  14  A.  3  R.  to  rods.  Ans.  59  rods. 

31.  Reduce  1  m.  2  fur.  to  yds.          Ans.  2200  yds. 

32.  Reduce  2  m.  3  fur.  to  ft.  Ans.  12540  ft. 

33.  Reduce  8  ft.  10  in.  to  in.  Ans.  106  in. 

34.  Reduce  1  yd.  2  ft.  to  in.  Ans.  60  in. 

35.  Reduce  2  y.  2  ft.  8  in.  to  in.  Ans.  144  in. 

36.  Reduce  1  mile  2  fur.  to  feet.  *      Ans.  6600  ft. 

37.  Reduce  2  m.  3  fur.  5  rods  to  inches. 

Ans.  15147  in. 

38.  Reduce  5  bu.  2  pks.  1  qt.  to  qts.  Ans.  177  qts. 

39.  Reduce  10  A.  1  R.  to  yards.   Ans.  49610  yds. 

40.  Reduce  5  A.  1  R.  2  r.  to  rods,   Ans.  842  rods. 

41.  Reduce  7  A.  1  ft.  to  inches.  Ans.  43908624  in. 

42.  Reduce  8  S.  13°  to  seconds. 

Ans.  910800  seconds. 

43.  Reduce  2  S.  3°  to  minutes.          Ans.  3780  mi. 

44.  Reduce  1  S.  4°  2'  3"  to  sec.     Ans.  122523  sec. 

45.  Reduce  5°  2'  3"  to  sec.  Ans.  18125  sec. 

46.  Reduce  2  da.  5  hrs.  to  mi.  Ans.  3180  mi. 

47.  Reduce  360  dys.  5  hrs.  to  mi.  Ans.  518700  mi. 

48.  Reduce  5  hrs.  2  mi.  to  sec.       Ans.  18120  sec. 


REDUCTION   DESCENDING.  297 

49.  Reduce  12  hrs.  3  mi.  to  sec.     Am.  43380  sec. 

50.  Reduce  1  yr.  1  mo.  1  da.  to  mi. 

Ans.  132480  mi. 

51.  Reduce  $2  3  cents  to  mills.     Ans.  2030  mills. 

52.  Reduce  5  fur.  2  r.  10  ft.  to  in.    Ans.  40110  in. 

53.  Reduce  2  fur.  3  r.  8  ft.  to  in.      Am.  16530  in. 

54.  Reduce  1  fur.  4  r.  5  ft.  to  ft.  Am.  731  ft, 

55.  Reduce  1  mi.  2  fur.  3  r.  to  yds. 

Am.  2216£  yds. 

56.  Reduce  1  acre  2  R.  3  r.  to  rods. 

Am.  243  rods. 

57.  Reduce  2  acres  3  R.  4  r.  to  feet. 

Ans.  120879  ft. 

V 

58.  Reduce  1  Ib.  2  oz.  to  grains.  Ans.  6720  grains. 

59.  Reduce  2  Ibs.  4  oz.  3  cwt.  to  cwt. 

Am.  563  cwt. 

60.  Reduce  3  Ibs.  5  oz.  4  cwt.  to  grs. 

Am.  19776  grs. 

61.  Reduce  16  oz.  5  drm.  to  drms.  Ans.  161  dims. 

62.  Reduce  2  Tun  1  hhd.  to  gills. 

Am.  18144  gills. 

63.  Reduce  4  Tun  1  hhd.  2  galls,  to  galls. 

Am.  1073  galls. 

64.  Reduce  7  yds.  2  ft.  3  in.  to  inches. 

Am.  279  in. 

65.  Reduce  5  ft.  6  in.  to  in.  Am.  66  in. 

66.  Reduce  1  sq.  m.  5  a.  to  in. 

Am.  4045852800  in. 


298 


INTERMEDIATE  ARITHMETIC. 


LESSON  XXI. 


.    REDUCTION    ASCENDING. 


Reduce  2139  farthings  to  £,  etc. 

Divide  2139  by  4,  be- 
cause 4  farthings  make  1 
penny : 


4  )  2139 
534 


OPEEATION. 


4  )  2139 

12  )  534        -  3  far. 
20  )  44      —  6d. 
~~2£   —4s. 

Write  the  3  remaining  far- 
things a  little  to  the  right.  The  quotient  found  is 
pence.  Divide  the  pence  by  12,  because  12  pence 
make  1  shilling : 

12  )  534 

44  —  6 

Write  the  6  remaining  pence  a  little  to  the  right 
The  quotient  found  is  shillings.  Divide  the  shillings 
by  20,  because  20  shillings  make  l£. 

20  )  44 

~2~—  4 

Write  the  remaining  shillings  a  little  to  the  right. 
The  quotient  is  pounds  sterling.  The  last  quotient, 
together  with  the  several  remainders,  form  the  com- 
pound number 


From  which  we  derive  the  following 


REDUCTION   ASCENDING.  299 

KULE. 

Divide  the  given  number  by  the  number  of  units 
of  its  kind  required  to  make  one  of  the  next  higher 
denomination.  If  any  remainder  write  it  a  little  to 
the  right  of  the  quotient.  Divide  the  quotient  by  the 
number  of  units  of  its  kind  required  to  make  one  of 
the  next  higher  denomination.  If  any  remainder, 
write  it  a  little  to  the  right  of  the  quotient.  Continue 
step  by  step  in  this  way  until  the  highest  denomina- 
tion required  is  reached.  The  remainders  together 
with  the  last  quotient  is  the  compound  number  re- 
quired. 

1.  Reduce  61495  dwt.  to  Ibs.,  etc. 

Ans.  27  Ibs.  15  dwt.  to  dwt. 

2.  Reduce  2676  far.  to  £,  etc. 

Ans.  2£  15s.  9d.  to  far. 

3.  Reduce  19440  far.  to  £,  etc. 

Ans.  20£  5s.  to  far. 

4.  Reduce  19920  far.  to  £,  etc. 

Ans.  2 0£  15s.  to  far. 

5.  Reduce  821  far.  to  £,  etc. 

Ans.  17s.  5  far.  to  far. 

6.  Reduce  6794  grs.  to  Ibs.,  etc. 

Ans.  1  Ib.  2  oz.  3  dwt.  2  gr. 

7.  Reduce  7320  grs.  to  Ibs.,  etc. 

Ans.  1  Ib,  3  oz.  5  dwt.  to  grs. 

8.  Reduce  1560  grs.  to  oz.,  etc. 

Ans.  3  oz.  5  dwt.  to  errs. 


300 


INTERMEDIATE   ARITHMETIC. 


9.  Reduce  224005  oz.  to  Tons,  etc. 

Ans.  7  Ton,  5  oz.  to  oz. 

10.  Reduce  544015  oz.  to  Tons,  etc. 

Ans.  17  Ton,  15  oz.  to  oz. 

11.  Reduce  428400  oz.  to  Tons,  etc. 

Ans.  13  Ton,  7  cwt.  3  qr.  to  oz. 

12.  Reduce  1110  quarters  to  Tons,  etc. 

Ans.  13  Tons,  17  cwt.  2  qr.  to  oz. 

13.  Reduce  6563  drms.  to  quarters,  etc. 

Ans.  1  qr.  10  oz.  3  dr.  to  dr. 

14.  Reduce  1128  in.  to  rods,  etc. 

Ans.  5r.  3  yds.  2  ft.  6  in. 

15.  Reduce  878  in  to  rods,  etc. 

Ans.  4r.  2  yds.  1  ft.  2  in.  to  in. 

16.  Reduce  14630  ft.  to  fur.,  etc. 

Ans.  22  fur.  6  r.  3  yds.  2  ft.  to  ft. 

17.  Reduce  101656  in.  to  miles,  etc. 

Ans.  4  m.  6  fur.  20  r.  16  in.  to  in. 

18.  Reduce  85i-  in.  to  yds.,  etc. 

Ans.  2  yds.  1  qr.  2  na.  to  inches. 

19.  Reduce  54  in.  to  ft.,  etc. 

Ans.  4  ft.  6  in.  to  inches. 

20.  Reduce  1013  qts.  to  Tuns,  etc. 

Ans.  1  Tun,  1  gal.  1  qt.  to  qt. 

21.  Reduce  16184  gills  to  Tuns,  etc. 

Ans.  2  Tun,  1  gall.  3  qts.  to  gills. 

22.  Reduce  24256  gills  to  Tuns,  etc, 

Ans.  3  Tuns,  2  galls,  to  gills. 

23.  Reduce  10224  pints  to  bbls.,  etc. 

Ans.  40  bbls.  18  galls,  to  pints. 


KEDUCTION   ASCENDING.  301 

24.  Reduce  506  quarts  to  bushels,  etc. 

Am.  15  bu.  2  pks.  to  quarts. 

25.  Reduce  620730  ft.  to  Acres,  etc.  , 

Am.  14  A.  1  R.  to  ft. 

26.  Reduce  68970  yds.  to  Acres,  etc. 

Ans.  14  A.  1  R.  to  yds. 

27.  Reduce  58  rods  to  acres,  etc. 

Ans.  14  A.  2  R.  to  rds. 

28.  Reduce  111417768  rods  to  acres,  etc. 

Am.  17  A.  3  R.  2  r.  to  in. 

29.  Reduce  4014489602  in.  to  miles,  etc. 

Ans.  10  miles  2  in.  to  in. 

30.  Reduce  59  rods  to  A.,  etc. 

Ans.  14  A.  3  R.  to  rods. 

31.  Reduce  2200  yds.  to  mi.,  etc. 

Ans.  1  mile  2  fur.  to  yds. 

32.  Reduce  12540  ft.  to  mi.,  etc. 

Ans.  2  miles  3  fur.  to  ft. 

33.  Reduce  106  in.  to  ft.,  etc. 

Ans.  8  ft.  10  in.  to  in. 

34.  Reduce  60  in.  to  yds.,  etc. 

Ans.  1  yd.  2  ft.  to  inches. 

35.  Reduce  144  to  yds.,  etc. 

Am.  2  yds.  2  ft.  8  in.  to  in. 

36.  Reduce  6600  to  miles,  etc. 

Ans.  1  m.  2  fur.  to  feet. 

37.  Reduce  15147  in.  to  mi.  etc. 

Am.  2  m.  3  fur.  5  r.  to  inches. 

38.  Reduce  177  qts.  to  bu.,  etc. 

Ans.  5  bu.  2  pks.  1  qt.  to  qts. 
26 


302 


INTEEMEDIATE   AEITHMETIC. 


39.  Reduce  49610  yds.  to  acres,  etc. 

Ans.  10  A.  1  R.  to  yards. 
£0.  Reduce  842  rods  to  A.,  etc. 

Ans.  5  A.  1  R.  2  r.  to  rods. 

41.  Reduce  43908624  in.  to  A.,  etc. 

Ans.  V  A.  0  R.  1  ft.  to  inches. 

42.  Reduce  910800  sec.  to  S.,  etc. 

Ans.  8  S.  13°  to  seconds. 

43.  Reduce  3780  mi.  to  S.,  etc. 

Ans.  2  S.  3°  to  minutes. 

44.  Reduce  122523  sec.  to  S.  etc. 

Ans.  1  S.  4°  2'  3"  to  sec. 

45.  Reduce  18125  sec.  to  deg.,  etc. 

Ans.  5°  2'  3"  to  sec. 

46.  Reduce  3180  mi.  to  da.,  etc. 

Ans.  2  days,  5  hours,  to  min. 

47.  Reduce  518700  mi.  to  da.,  etc. 

Ans.  360  days,  5  hrs.  to  min. 

48.  Reduce  18120  sec.  to  da.,  etc. 

Ans.  5  hours,  2  min.  to  sec. 

49.  Reduce  43380  sec.  to  hrs.,  etc. 

Ans.  12  h.  3  min.  to  sec. 

50.  Reduce  132480  mi.  to  yds.,  etc. 

Ans.  1  yr.  1  m.  1  da.  to  min. 

51.  Reduce  2030  mills  to  dollars,  etc. 

Ans.  $2,  3  cts.  to  mills. 

52.  Reduce  40116  in.  to  fur.,  etc. 

Ans.  5  fur.  2  r.  10  ft.  to  in. 

53.  Reduce  16530  in.  to  fur.,  etc. 

Ans.  2  fur.  3  r.  8  ft.  to  in. 


REDUCTION   ASCENDING.  303 

54.  Reduce  731  ft.  to  fur.,  etc. 

Ans.  1  fur.  4  r.  5  ft.  to  ft. 

55.  Reduce  2216£  yds.  to  miles,  etc. 

Ans.  1  m.  2  fur.  3  r.  to  yds. 

56.  Reduce  243  rods  to  A.,  etc. 

Ans.  1  A.  2  R,  3  r.  to  r. 

57.  Reduce  120879  ft.  to  A.,  etc. 

Ans.  2  A.  3  R.  4  r.  to  ft. 

58.  Reduce  6720  grains  to  Ibs.,  etc. 

Ans.  1  Ib.  2  oz.  to  grains. 

59.  Reduce  563  dwt.  to  Ibs.,  etc. 

Ans.  2  Ib.  4  oz.  3  dwt.  to  dwt. 

60.  Reduce  19776  grs.  to  Ibs.,  etc. 

Ans.  3  Ib.  5  oz.  4  dwt.  to  gr. 

61.  Reduce  161  drams  to  oz.,  etc. 

Ans.  16  oz.  5  dr.  to  drams. 

62.  Reduce  18144  gills  to  hhd.,  etc. 

Ans.  2  tuns,  Ihhd.  to  gills. 

63.  Reduce  1073  galls,  to  Tuns,  etc. 

Ans.  4  tuns,  1  hhd.  4  gal.  to  gal. 

64.  Reduce  279  in.  to  yards,  etc. 

Ans.  7  yds.  2  ft.  3  in.  to  in. 

65.  Reduce  66  in.  to  ft.,  etc. 

Ans.  5  ft.  6  in.  to  inches- 

66.  Reduce  4045852800  in.  to  sq.  m.,  etc. 

Ans.  1  sq.  m.  5  A.  to  in. 


304  INTERMEDIATE   ARITHMETIC. 


LESSOR  XXII. 

334.    REDUCTION  OF  DENOMINATE  FRACTIONS  TO  COM- 
POUND   NUMBERS. 

EXAMPLE. 

Reduce  f£  to  a  compound  number.  Ans.  13s.  4d. 

Reduce  f  of  a  pound  to  shillings  by  multiplying 
by  20,  because  20  shillings  make  l£. 

I  X  \°-  =  -4/  =  13|  shillings.  Reduce  £  of  a  shil- 
ling to  pence  by  multiplying  by  12,  because  12  pence 
make  1  shilling.  |  x  ^  =  -V2-  =  4  pence.  The  an- 
swer is  13s.  4d. 

RULE. 

Reduce  the  given  fraction  to  the  next  lower  de- 
nomination. If  the  product  ends  in  a  fraction^  reduce 
the  fraction  to  the  next  lower  denomination.  If  the 
Id  product  ends  in  a  fraction^  reduce  that  fraction  to 
the  next  lower  denomination.  Proceed  in  this  way 
until  the  lowest  unit  of  the,  table  is  reached.  If  there 
be  any  remainder,  place  it  in  the  form  of  a  fraction. 
The  several  integral  parts  of  the  product  will  be  the 
required  compound  number. 

1.  Reduce  f  £  to  a  compound  number. 

Ans.  13s.  4d.  to  a  denominate. 

2.  Reduce  $  £  to  a  compound  number. 

Ans.  14s.  3d.  7f  far. 

3.  Reduce  f  £  to  a  compound  number. 

Ans.  17s.  Id.  2f  far. 


REDUCTION    OF   DENOMINATE   FRACTIONS.         305 

4.  Reduce  |  £  to  a  compound  number. 

Am.  13s.  4d. 

5.  Reduce  \  £  to  a  compound  number. 

Ans.  2s.  2d.  2~  far. 

6.  Reduce  £  £  to  a  compound  number.  ^«s.  10s. 

7.  Reduce  -J-  s.  to  a  compound  number. 

Ans.  3s.  4d. 

8.  Reduce  -J-  Ib.  Troy  to  a  compound  number. 

^IMS.  1  oz.  10  dwt. 

9.  Reduce  £  Ib.  Troy  to  a  compound  number. 

Ans.  2  oz.  13  d\vt.  8  grs. 

10.  Reduce  |-  Ib.  Troy  to  a  compound  number. 

Ans.  10  oz. 

11.  Reduce  ^  oz.  Troy  to  a  compound  number. 

Ans.  2  dwt.  10  grs. 

12.  Reduce  -f  oz.  Troy  to  a  compound  number. 

Ans.  4  dwt.  10  j  grs. 

13.  Reduce  -ff  oz.  Troy  to  a  compound  number. 

Ans.  10  dwt.  21T\  grs. 

14.  Reduce  f  Ton  Av.  to  a  compound  number. 

Ans.  6  cwt. 

15.  Reduce  T1T  Ton  Av.  to  a  compound  number. 

Ans.  I  cwt.  1  qr.  17  Ibs.  13  oz.  11-f-  dr. 

16.  Reduce  %  cwt.  Av.  to  a  compound  number. 

Ans.  1  qr.  8  Ibs.  5  oz.  5*  drs. 

17.  Reduce  f  cwt.  Av.  to  a  compound  number. 

Ans.  2  qr.  16  Ibs.  10  oz.  lOf  drs. 

18.  Reduce  £  qr.  Av.  to  a  compound  number. 

Ans.  20  Ibs.  13  oz.  5-J-  drs. 

19.  Reduce  ^  qr.  Av.  to  a  compound  number. 

Ans.  17  Ibs.  13  oz.  llf  drs. 
26* 


306 


INTERMEDIATE   AKITHMETTC. 


20.  Reduce  ^  Ib.  Av.  to  a  compound  number. 

Ans.  2  oz. 

21.  Reduce  |  Ib.  Av.  to  a  compound  number. 

Ans.  10  oz. 

22.  Reduce  -J-  3  to  a  compound  number. 

Ans.  1310  grs. 

23.  Reduce  •£  3  to  a  compound  number. 

^iws.  4  grs. 

24.  Reduce  J  mile  to  a  compound  number. 

Ans.  2  fur.  26  r.  3  y.  2  ft. 

25.  Reduce  -|  mile  to  a  compound  number. 

-4»*.  6  fur.  26  r.  3  y.  2  ft. 

26.  Reduce  £  fur.  to  a  compound  number. 

Am.  20  rods. 

27.  Reduce  -f  fur.  to  a  compound  number. 

Ans.  33  r.  1  yd.  2  ft.  6  in 

28.  Reduce  ^  rod  to  a  compound  number. 

Ans.  1  yd.  2  ft.  6  in. 

29.  Reduce  %  yard  to  a  compound  number. 

Ans.  1  ft.  6  in. 

30.  Reduce  f  foot  to  a  compound  number. 

-4«s.  7i  inches. 

31.  Reduce  ^  league  to  a  compound  number. 

Ans.  1  fur.  8  rods. 

32.  Reduce  £  foot  to  a  compound  number. 

Ans.  6  inches. 

33.  Reduce  -^  E.  F.  to  a  compound  number. 

Ans.  I  qr. 

34.  Reduce  \  E.  E.  to  a  compound  number. 

Ans.  1  qr. 


EEDUCTION   OF   DENOMINATE  FEACTIONS.         307 

35.  Reduce  J  E.  Fl.  to  a  compound  number. 

Ans.  I  qr. 

36.  Reduce  J  yd.  to  a  compound  number. 

Ans.  3  qr. 

37.  Reduce  f  A.  to  a  compound  number. 

Ans.  3  R.  13  r.  10  y.  0  ft.  108  in. 

38.  Reduce  f  R.  to  a  compound  number. 

Ans.  26  r.  20  yds.  1  ft.  72  in. 

39.  Reduce  %  S.  rod  to  a  compound  number. 

Ans.  25  yds.  1  ft.  126  in. 

40.  Reduce  %  sq.  yd.  to  a  compound  number. 

Ans.  4  ft.  72  inches. 

41.  Reduce  j-  sq.  ft.  to  a  compound  number. 

Ans.  36  sq.  in. 

42.  Reduce  |-  sq.  ft.  to  a  compound  number. 

Ans.  120  sq.  in. 

43.  Reduce  -J  sq.  yd.  to  a  compound  number. 

Ans.  7  sq.  ft.  126  in. 

44.  Reduce  |  sq.  rod  to  a  compound  number. 

Ans.  16  sq.  yds.  7  ft.  36  in. 

45.  Reduce  f  R.  to  a  compound  number. 

Ans.  24  rods. 

46.  Reduce  J-  A.  to  a  compound  number. 

Ans.  17  r.  23  yds.  4  ft.  108  in. 

47.  Reduce  •£%  sq.  m.  to  a  compound  number. 

Ans.  26  r.  20  y.  1  ft.  48  in. 

48.  Reduce  \  cu.  ft.  to  a  compound  number. 

Ans.  864  cubic  inches. 

49.  Reduce  f  cu.  ft.  to  a  compound  number. 

Ans.  1152  cu.  inches. 


308  INTEKMEDIATE   AEITHMETIC. 

50.  Reduce  -f  tun  to  a  compound  number. 

Am.  1  pi.  1  hhd.  12  gal.  2  qt.  f  pts. 

51.  Reduce  ^  pi.  to  a  compound  number. 

A ns.  42  galls. 

52.  Reduce  -f-  hhd.  to  a  compound  number. 

Ans.  45  gall. 

53.  Reduce  -f  gill  to  a  compound  number. 

Ans.  1  qt.  1  pt.  f  gill. 

54.  Reduce  -J  qrt.  to  a  compound  number. 

Ans.  2f  gills. 

55.  Reduce  f  pi.  to  a  compound  number. 

4«*.  1  hhd.  21  galls. 

56.  Reduce  \  bbl.  to  a  compound  number. 

Ans.  15  gall.  3  qt. 

57.  Reduce  ^  pi.  to  a  compound  number. 

-4»s.  10  gal.  2  qt. 

58.  Reduce  ^  qr.  to  a  compound  number. 

Ans.  2  bu.  5  qt.  f  pts. 

59.  Reduce  -J-  bu.  to  a  compound  number. 

Ans.  6  qts.  -|  pt. 

60.  Reduce  |-  pk.  to  a  compound  number. 

Ans.  1  qt.  -f  pts. 

61.  Reduce  |  qt.  to  a  compound  number. 

Ans.  |-  pt. 

62.  Reduce  |-  yr.  to  a  compound  number. 

Ans.  182  da.  12  hrs. 

63.  Reduce  f  yr.  to  a  compound  number. 

Ans.  243  da.  8  hours. 

64.  Reduce  ^  da.  to  a  compound  number. 

Ans.  4  h.  48  mint 

65.  Reduce  ^  S.  to  a  compound  number. 

Ans.  3°  20'. 


REDUCTION    OF   DENOMINATE   NUMBERS.  309 


LESSON     XXIII. 

325.     REDUCTION      OF      DENOMINATE      NUMBERS      TO 
COMPOUND    FRACTIONS. 

EXAMPLE. 

1.  Reduce  13s.  4d. to  a  compound  fraction. 

Reduce  13s.  4d.  to  pence.  13x12  =  156  pence, 
156+4=160  pence.  Reduce  1  £.  to  pence,  20  x  12= 
240  pence.  Make  the  given  number  (reduced  to 
pence)  the  numerator  and  the  £  reduced  to  pence 
the  denominator.  We  now  have  Jff  for  our  frac- 
tion. Reduced  to  its  lowest  terms  gives  f  £  for  the 
answer. 

KULE. 

Reduce  the  unit  of  the  proposed  fraction  to  thJ 
lowest  denomination  of  the  given  denominate  number. 
Make  this  the  Denominator. 

Reduce  the  given  compound  number  to  the  same 
unit.  Make  this  the  Numerator. 

Reduce  the  fraction  to  its  lowest  terms. 

1.  Reduce  13s.  4d.  to  the  fraction  of  a  £. 

Ans.  |£. 

2.  Reduce  14s.  3d.  If  far.  to  the  fraction  of  a  £. 

Am.  |£. 

3.  Reduce  17s.  Id.  2$  far.  to  the  fraction  of  a  £. 

Ans.  f  £. 

4.  Reduce  13s.  4 d.  to  the  fraction  of  a  £. 

Ans.  |£. 


310 


INTERMEDIATE   ARITHMETIC. 


5.  Reduce. 2s.  2d.  2|  far.  to  the  fraction  of  a  £. 

Am.  i£. 

6.  Reduce  10s.  to  the  fraction  of  a  £.     Am.  |£. 

7.  Reduce  3s.  4d.  to  the  fraction  of  a  £. 

Am.  Js. 

8.  Reduce   1  oz.  10  dwt.  Troy  to  the  fraction 
of  a  Ib.  Am.  %  Ib. 

9.  Reduce  2  oz.  13  dwt.  8  grs.  Troy  to  the  frac- 
tion of  a  Ib.  Am.  f  Ib. 

10.  Reduce  10  oz.  Troy  to  the  fraction  of  a  Ib. 

Am.  I  Ib. 

11.  Reduce  2  dwt.  12  grs.  Troy  to  the  fraction  of 
an  oz.  Am.  -J-  oz. 

12.  Reduce  4  dwt.  lOf  grs.  Troy  to  the  fraction 
of  an  oz.  Ans.  |  oz. 

13.  Reduce  10  dwt.  21^  grs.  Troy  to  the  fraction 
of  an  oz.  Ans.  T6T  oz. 

14.  Reduce  6  cwt.  to  the  fraction  of  a  ton. 

Ans.  f  ton. 

15.  Reduce  1  cwt.  1  qr.  17  Ibs.  13  oz.  llf  dr.  to 
the  fraction  of  a  ton.  Am.  -^  ton. 

16.  Reduce  1  qr.  8  Ibs.  5  oz.  5-£-  dr.  to  the  fraction 
of  a  cwt.  A.ns.  -|-  cwt. 

17.  Reduce  2  qrs.  16  Ibs.  10  oz.  lOf  dr.  to  the  frac- 
tion of  a  cwt.  Ans.  f  cwt. 

18.  Reduce  20  Ibs.  13  oz.  51  drs.  to  the  fraction 
of  a  qr.  Ans.  -f  qr. 

19.  Reduce  17  Ibs.  13  oz.  llf  drms.  to  the  frac- 
tion of  a  qr.  Am.  -§-  qr. 

20.  Reduce  2  oz.  to  the  fraction  of  a  Ib. 

Ant.  -   Ib. 


KEDUCTIOX   OF   DENOMINATE  NUMBEES.          311 

21.  Reduce  10  oz.  to  the  fraction  of  a  Ib. 

Ans.  -«  H). 

22.  Reduce  1  3  10  grs.  to  the  fraction  of  a  3  . 

Ans.  £  oz. 

23.  Reduce  4  grs.  to  the  fraction  of  a  3. 


24.  Reduce  2  fur.  2G  r.  3  yds.  2  It.  to  the  fraction 
of  a  mile.  Ans.  %  m. 

25.  Reduce  6  fur.  26  r.  3  y.  2  ft.  to  the  fraction 
of  a  mile.  Ans.  £  m. 

26.  Reduce  20  rods  to  the  fraction  of  a  fur. 

Ans.  £  fur. 

27.  Reduce  33  r.  1  yd.  2  ft.  6  in.  to  the  fraction 
of  a  fur.  Ans.  {:  fur. 

28.  Reduce   1   yd.  2  ft.  6  in.  to  the  fraction  of  a 
rod.  Ans.  -J-  rod. 

29.  Reduce  1  ft.  6  in.  to  the  fraction  of  a  yd. 

Ans.  %  yd. 

30.  Reduce  7|  inches  to  the  fraction  of  a  foot. 

Ans.  -f  ft. 

31.  Reduce  1  fur.  8  rods  to  the  fraction   of  a 
league.  Ans.  ^V  lea. 

32.  Reduce  6  in.  to  the  fraction  of  a  foot. 

Ans.  %  ft. 

33.  Reduce  1  qr.  Cloth  Measure  to  the  fraction 
of  a  E.  E.  Ans.  %  E.  E. 

34.  Reduce  1  qr.  to  the  fraction  of  a  E.  F. 

Ans.  -tE.F. 

35.  Reduce  1  qr.  to  the  fraction  of  a  E.  F. 

Ans.     E.  F. 


312 


INTERMEDIATE   ARITHMETIC. 


36.  Reduce  3  qrs.  to  the  fraction  of  a  yd. 

Ans. 

37.  Reduce  3  R.    13  r.   10  y.   0  ft.  108  in.  to  the 
fraction  of  an  acre.  Ans.  -f-  acre. 

38.  Reduce  26  r.  20  y.  1  ft.  72  in.  to  the  fraction 
of  a  rood.  Ans.  f  rood- 

39.  Reduce  25  y.  1  ft.  126  in.  to  the  fraction  of  a 
rod.  Ans.  |  rod. 

40.  Reduce  4  ft.  72  in.  to  the  fraction  of  a  yard. 

Ans.  \  sq.  yd. 

41.  Reduce  36  sq.  in.  to  the  fraction  of  a  foot. 

Ans.  J  sq.  ft. 

42.  Reduce  1 20  sq.  in.  to  the  fraction  of  a  foot. 

Ans.  -f  sq.  ft. 

43.  Reduce  7  sq.  ft.  126  in.  to  the  fraction  of  a 
yard.  Ans.  -J  sq.  yd. 

44.  Reduce  16  sq  yd.  7  ft.  36  in.  to  the  fraction  of 
a  rod.  Ans.  -f  sq.  rod. 

45.  Reduce  24  r.  to  the  fraction  of  a  rood. 

Ans.  |  rood. 

46.  Reduce  17  rods  23  yds.   4  ft.   108  in.  to  the 
fraction  of  an  acre.  Ans.  £  acre. 

47.  Reduce  426  A.  2  R.  26  r.  20  y.  1  ft.  48  in.  to 
the  fraction  of  a  mile.  Ans.  -^  sq.  mile. 

48.  Reduce  864  cubic  inches  to  the  fraction  of  a 
cu.  ft.  Ans.  %  cu.  ft. 

49.  Reduce    1152  cubic  in.  to  the  fraction  of  a 
cu.  ft.  Ans.  |  cu.  ft. 

50.  Reduce  1  pi.  1  hhd.  12  galls.  3  qts.  -f  pint  to 
the  fraction  of  a  tun.  Ans.     tun. 


REDUCTION   OF   DENOMINATE   Nl'MDEKS.  313 

51.  Reduce  42  gallons  to  the  fraction  of  a  hhd. 

Ans.  ^  pipe. 

52.  Reduce  45  galls,  to  the  fraction  ol  a  lilid. 

Ans.  %  hhd. 

53.  Reduce  1  qt.  1  pt.  %  gill  to  the  fraction  of  a 
gall.  Ans.  |  gall. 

54.  Reduce  2|  gills  to  the  fraction  of  a  quart. 

Ans.  $  quart. 

55.  Reduce  1  hhd.  21  galls,  to  the  fraction  of  a 
pipe.  Ans.  -J  pipe. 

56.  Reduce  15  galls.  3  qts.  to  the  fraction  of  a 
bbl.  Ans.  £  bbl. 

57.  Reduce  10  galls.  2  qts.  to  the  fraction   of  a 
tierce.  Ans.  £  tierce. 

58.  Reduce  2  bu.  5  qts.  f  pt.  to  the  fraction  of  a 
qr.  Ans.  J  qr. 

59.  Reduce  6  qts.  f  pt.  to  the  fraction  of  a  bu. 

Ans.  %  bu. 

60.  Reduce  1  qt.  f  pt.  to  the  fraction  of  a  pk. 

Ans.  %  pk. 

61.  Reduce  f  pt.  to  the  fraction  of  a  qt. 

Ans.  I  qt. 

62.  Reduce  182  days  12  hrs.  to  the  fraction  of  a  yr. 

Ans.  ^  yr. 

63.  Reduce    243  days   8  hrs.  to  the  fraction  of 
a  yr.  Ans.  f  yr. 

64.  Reduce  4  hrs.  48  mi,  to  the  fraction  of  a  da. 

Ans.  ^  da. 

65.  Reduce  3°  20'  to  the  fraction  of  a  S. 

Ans.  %  S. 
2V 


314 


INTERMEDIATE  ARITHMETIC. 


LESSON  XXIY. 

336.   REDUCTION    OF   COMPOUND   NUMBERS  TO 
DECIMALS. 

CASE   I. 
EXAMPLE. 

Reduce  1  gill  to  the  decimal  of  a  gallon. 

Reduce  1  gallon  to  gills.  Make  the  product  the 
denominator.  Make  the  given  number  the  numer- 
ator. Thus  :  In  1  gallon  there  are  4  X  2  X  4  =  32 
gills. 

The  fraction  is  ^  reduced  to  a  decimal,  gives 
,03125  for  the  answer. 

Or: 

Reduce  gills  to  the  decimal  of  a  pint  by  dividing 
by  4,  which  gives  .25   pints.     Divide        OPERATION. 
.25  pints  by  2,  to  reduce  it  to  the  de- 
cimal  of   a   quart.     This    gives   .125 
quarts,  which,  divided  by  4,  reduces       2)0.25 
it  to  the  decimal  of  a  gallon,  the  an- 
swer required.     Ans.  0.03125.  4)0.125 


4)1. 


0.03125 

EXAMPLES. 

1.  Reduce  1  foot  to  the  decimal  of  a  mile. 

Ans.  .0001 89-Jf. 

2.  Reduce  1  gill  to  the  decimal  of  a  gallon. 

Ans.  .03125  gills. 

3.  Reduce  2  oz.  to  the  decimal  of  a  Ton. 

Am.  .0000625  Ton. 


REDUCTION   OF   COMPOUND   NUMBERS.  315 

4.  Reduce  12  Ibs.  to  the  decimal  of  a  Ton. 

Am.  .006  TOD. 

5.  Reduce  1  far.  to  the  decimal  of  a  £. 

Ans.  .00104^  £. 

6.  Reduce  3  far.  to  the  decimal  of  a  £. 

Ans.  .003125  £. 

7.  Reduce  1  pint  to  the  decimal  of  a  hhd. 

Ans.  .00 1984-^- hhd. 

8.  Reduce  18  pounds  to  the  decimal  of  a  cwt. 

Ans.  .18  cwt. 

9.  Reduce  23  pounds  to  the  decimal  of  a  Ton. 

Ans.  .0115  Ton. 

10.  Reduce  9d.  to  the  decimal  of  a  £. 

Ans.  .0375  £. 

11.  Reduce  15s.  to  the  decimal  of  a  £. 

Ans.  .75  £. 

12.  Reduce  3  qts.  to  the  decimal  of  a  Tun. 

Ans.  .0029J-J  Tun. 

13.  Reduce  2  qrs.  to  the  decimal  of  a  Ton. 

Ans.  .025  Ton. 

14.  Reduce  2  qrs.  to  the  decimal  of  a  cwt. 

Ans.  .5  cwt. 

15.  Reduce  3  qrs.  to  the  decimal  of  a  Ton. 

Ans.  .0375  Ton. 

16.  Reduce  6  oz.  to  the  decimal  of  a  Ton. 

Ans.  .0001875  Ton. 

17.  Reduce  1  sh.  to  the  decimal  of  a  Cr. 

Ans.  .2  Cr. 

18.  Reduce  1  gr.  to  the  decimal  of  a  Ib. 

Ans.  .000174-3^  Ib. 


,316  INTERMEDIATE   ARITHMETIC. 

19.  Reduce  1  dr.  to  the  decimal  of  a  Ton. 

Ans.  .000001953125  Ton. 

20.  Reduce  1  in.  to  the  decimal  of  a  ft. 

Ans.  .0833J  ft. 

21.  Reduce  2  in.  to  the  decimal  of  a  yd. 

Ans.  .0555|-  yd. 

22.  Reduce  3  in.  to  the  decimal  of  a  rod. 

Ans.  .0151JJ  rod. 

23.  Reduce  4  in,  to  the  decimal  of  a  fur. 

Ans.  .OOOSOSj-Lg-  fur. 

24.  Reduce  5  in.  to  the  decimal  of  a  mile. 

Ans.  .000078125  m. 

25.  Reduce  6  in.  to  the  decimal  of  a  Lea. 

Ans.  .000031 5 Jl-jj-  Lea. 

26.  Reduce  1  Rood  to  the  decimal  of  a  sq.  mile. 

Ans.  .0039-Jg-  sq.  m. 

27.  Reduce  9  sq.  ft.  to  the  decimal  of  a  Rood. 

Ans.  .00082644^  R. 

28.  Reduce  2  pints  to  the  decimal  of  a  pipe. 

Ans.  .001 9ff  pipe. 

29.  Reduce  2  galls,  to  the  decimal  of  a  Tun. 

Ans.  .0079f|  Tun. 

30.  Reduce  40  galls,  to  the  decimal  of  a  hhd. 

Ans.  6349i|  hhd. 

31.  Reduce  1  gill  to  the  decimal  of  a  Tun. 

Ans.  .0001 24^  Tun. 

32.  Reduce  2  qrts.  to  the  decimal  of  a  Tun. 

Ans.  .0198fVg-Tun. 

33.  Reduce  20£.   15s.  9d.  3  far.  to  the   decimal 
ofa£.  Ans.  20.790625£= 


DEDUCTION    OF    COMPOUND    NTMUKKS.  317 

34.  Reduce  1  hhd.  25  galls.  3  qts.  to  the  decimal 
ofahhd.  Ans.  1.410714T2T  hhd. 

35.  Reduce  23  cwt.  2  qrs.  18  Ibs.  to  the  decimal  of 
a  Ton.  Ans.  1.184  Ton. 

3C.  Reduce  9  cwt.  3  qrs.  23  Ibs.  to  the  decimal  of 
a  Ton.  Ans.  .490  Ton. 

37.  Reduce  14£  4s.  Cd.  2  far.  to  the  decimal  of  a 
£.  Ans.  14.22708J  £. 

38.  Reduce  1  Ib.  9  oz.  13  dwt.  to  the  decimal  of 
a  Ton.  Ans.  .00080G640625  Ton. 

39.  Reduce  19  cwt.  3  qrs.  23  Ibs.  to  the  decimal 
of  a  Ton.  Ans.  .999  Ton. 

40.  Reduce  18  galls.  3  qts.  1  pt.  to  the  decimal 
of  a  hhd.  Ans.  .2990/j  hhd. 

41.  Reduce  365  d.  4  hrs.  48  min.  to  the  decimal 
of  a  year,  Ans.  1.000|-|f;  yr. 

42.  Reduce  4  A.  3  R.  20  r.  to  the  decimal  of  a 
mile.  Ans.  .7017  -     mile. 


LESSON  XXY. 

CASE  II. 

337.  Reduce  20  £  15s.  9d.  3  farthings  to  the 
decimal  of  a  £. 

Find  how  many  farthings  in  the  given  number, 
leaving  out  the  20  £  which  will  be  a  whole  number. 
27* 


318  INTERMEDIATE    ARITHMETIC. 

15s.  x  12  =  180  pence,  180  +  9=189  pence. 
189d.  x  4  =  756  far.  756  +  3  =  759  farthings. 

"Now  find  how  many  farthings  in  a  £  and  make 
the  result  the  denominator  of  a  fraction  having  the 
first  result  for  its  numerator. 

20  x  12  x  4  =  960  farthings  in  1  £. 

The  fraction  will  be  J-jjjj.,  which  reduced  to  a  dec- 
imal gives  0.790625.  Write  the  whole  number  (20  £) 
before  the  decimal  point  and  you  get  the  required 
answer.  Ans.  20.790625. 

Or: 

OPERATION. 
4)3 


12)9.75 
20)15.8125 


20.790625 


Divide  the  lowest  de- 
nomination,  farthings,  by  the  num- 
ber  required  to  make  one  of  a 
higher  denomination.  This  gives 
0.75  pence.  Write  the  number  of 
pence  to  the  left  of  the  decimal 
and  divide  by  the  number  of  pence  required  to  make 
one  of  the  next  higher  denomination.  This  gives 
0.8125  shillings.  Write  the  given  number  of  shil- 
lings to  the  left  of  the  decimal  point  and  divide  by 
the  number  of  shillings  required  to  make  one  pound 
sterling.  This  gives  0.790625.  Write  the  number 
of  £  to  the  left  of  the  decimal  and  you  have  the  re- 
quired answer. 


REDUCTION   OF   COMPOUND   NUMBERS.  3]  9 

339.  When  reducing  long  measure  or  square 
measure,  the  pupil  will  be  careful  and  remember  that 
if  there  is  a  remainder  after  dividing  by  5|-  or  30J-, 
that  remainder  is  a  fractional  part  of  a  rod  and  should 
be  immediately  reduced  to  yards. 

43.  Reduce  2  cwt.  1  qr.  6  Ibs.  to  the  decimal  of  a 
cwt.  Ans,  2.31  cwt. 

44.  Reduce  21  miles  to  the  decimal  of  a  degree. 

Ans.  .3036Jf  deg. 

45.  Reduce  1  sq.  rod  57  ft.  to  the  decimal  of  a 
rood.  Ans.  .0309  roods. 

46.  Reduce  13  £  5s.  to  the  decimal  of  a  guinea. 

Ans.  12.619^  G. 

47.  Reduce  14s.  9d.  to  the  decimal  of  a  £. 

Ans.  .737-5  £. 

48.  Reduce  17  cwt.  3  qr.  10  Ibs.  to  the  decimal  of 
a  ton.  Ans.  .8925  ton. 

49.  Reduce  10  £  14s.  2  far.  to  the  decimal  of  a  £. 

Ans.  10.720f  £. 

50.  Reduce  1  mi.  4  fur.  4  rods  to  the  decimal  of  a 
mile.  Ans.  1.5125  mile. 

51.  Reduce  5  A.  3  R.  20  r.  to  the  decimal  of  an 
acre.  Ans..  5.875  acre. 

52.  Reduce  8  cwt.  3  qrs.  6  Ibs.  to  the  decimal  of 
a  ton.  Ans.  .4405  ton. 


320  INTERMEDIATE   ARITHMETIC. 

53.  Reduce  3  qrs.  6  Ibs.  to  the  decimal  of  a  cwt. 

Ans.  .81  cwt. 

54.  Reduce  8  oz.  2  dwt.  5  grs.  to  the  decimal  of 
a  Ib.  Ans.  .67686||  Ib. 

55.  Reduce  2  yds.  3  na.  to  the  decimal  of  a  yd. 

Ans.  2.1875  yds. 

56.  Reduce  2°  15'  15"  to  the  decimal  of  a  deg. 

Ans.  .25416f  deg. 

57.  Reduce  20  Ibs.  10  dwt.  to  the  decimal  of  a  Ib. 

Ans.  20.4 16|  Ib. 

58.  Reduce  255   da.  16  hrs.  2  mi.  10  sec.  to  the 
decimal  of  a  yr.  Ans.  -1^^-fff^  yr. 

59.  Reduce  94  £  7s.  8d.  to  the  decimal  of  a  £. 

Ans.  94.3831  £. 

60.  Reduce  2  £  12s.  8d.  to  the  decimal  of  a  £. 

Ans.  2.631  £. 

61.  Reduce  2  bu.  1  pk.  7  qts.  to  the  decimal  of  a 
bushel.  Ans.  2.46875  bu. 

62.  Reduce  10  ton  5  cwt.  2  qrs.  18  Ibs.  to  the 
decimal  of  a  ton.  Ans.  10.284  ton. 

63.  Reduce  84°  18' 15"  to  the  decimal  of  a  deg. 

Ans.  84.304016f  deg. 

64.  Reduce  84°  18'  15"  to  the  decimal  of  a  circ. 

Ans.  .234 17^^  cir. 

65.  Reduce  4°  18'  15"  to  the  decimal  of  a  circ. 

Ans.  .0119ff~|-  circum. 


REDUCTION   OF   DENOMINATE   DECIMALS. 


321 


LESSON  XXVI. 

3J.O.    REDUCTION  OF  DENOMINATE  DECIMALS  TO  COM- 
POUND  NUMBEKS. 

EXAMPLES. 

1.  Reduce  0.790625  £  to  a  compound  number. 


Multiply  by  20  thus,  and  cut  off 
the  required  number  of  decimals. 
This  will  give  shillings  and  a  fraction. 
Multiply  the  fraction  by  12  and  cut 
off  the  required  number  ot  decimals. 
This  will  give  pence  and  a  fraction. 
Multiply  the  fraction  by  4  and  cut  off 
the  required  number  of  decimals. 
This  gives  3  farthings  and  no  remain- 
der. The  answer  is  then,  15s.  9d. 
3  far. 


OPERATION. 

.790625 
20 

15.812500 
12 


9.7500 
4 

3.00 


KULE. 

Multiply  the  given  decimal  fraction  by  the  number 
representing  the  next  lower  denomination.  Cut  off 
the  required  number  of  decimals.  Multiply  the  deci- 
mal part  by  the  figure  representing  the  next  lower  de- 
nomination^ cut  off  the  required  number  of  decimals 
and  continue  till  the  lowest  unit  is  reached.  The  inte- 
grals of  the  products  form  the  answer. 


322  INTERMEDIATE   ARITHMETIC. 

EXAMPLES. 

1.  Reduce  .000189^4  miles  to  a  compound  num- 
ber. Ans.  I  foot. 

2.  Reduce  .03125   gallon  to  a  compound  num- 
ber. Ans.  1  gill. 

3.  Reduce  .0000325  Ton  to  a  compound  num- 
ber. Ans.  2  oz. 

4.  Reduce  .006  Ton  to  a  compound  number. 

Ans.  12  Ib. 

5.  Reduce  .00104J-  £  to  a  compound  number. 

Ans.  I  far. 

6.  Reduce  .003125  £  to  a  compound  number. 

Ans.  3  far. 

7.  Reduce  .001984/3  hhd.  to  a  compound  num- 
ber. Ans.  1  pint. 

8.  Reduce  .18  cwt.  to  a  compound  number. 

Ans.  18  Ib. 

9.  Reduce  0.0115  Ton  to  a  compound  number. 

Ans.  23  Ib. 

10.  Reduce  .0375  £  to  a  compound  number. 

Ans.  9d. 

11.  Reduce  .75  £  to  a  compound  number. 

Ans.  15s. 

12.  Reduce  ,0029ff  Tun  to  a  compound  number. 

Ans.  3  qt. 

13.  Reduce  .025  Ton  to  a  compound  number. 

Ans.  2  qrs. 

14.  Reduce  .5  cwt.  to  a  compound  number. 

Ans.  2  qrs. 


REDUCTION    OF   DENOMINATE  DECIMALS.          323 

15.  Reduce  .0375  Ton  to  a  compound  number. 

Ans.  3  qrs. 

16.  Reduce  .0001875  Ton  to  a  compound  number. 

Ans.  6  oz. 

17.  Reduce  .2  Crown  to  a  compound  number. 

Ans.  Is. 

18.  Reduce  .000174^-  Ibs.  to  a  compound  num- 
ber. Ans.  1  qr. 

19.  Reduce  .0000001953125  Ton  to  a  compound 
number.  Ans.  1  dr. 

20.  Reduce  .083J  ft.  to  a  compound  number. 

Ans.  1  inch. 

21.  Reduce  0.0555  J  yd.  to  a  compound  number. 

Ans.  2  inches. 

22.  Reduce  0.0151  y  rod  to  a  compound  number. 

Ans.  3  inches. 

23.  Reduce  .OOOoOSy^  fur.  to  a  compound  num- 
ber. Ans.  4  inches. 

24.  Reduce  .00078125  mile  to  a  compound  num- 
ber. Ans.  5  inches. 

25.  Reduce  .0000315-f-jj-  Lea.  to  a  compound  num- 
ber. Ans.  6  inches. 

26.  Reduce  .0039^  sq.  m.  to   a  compound  num- 
ber. Ans.  1  Rood. 

27.  Reduce  .00082644-^:  R.  to  a  compound  num- 
ber. Ans.  9  sq.  yds. 

28.  Reduce  .0019|-J  pipe  to  a  compound  number. 

Ans.  2  pints. 

29.  Reduce  .0079||  Tun  to  a  compound  number. 

Ans.  2  galls. 


324 


IISTEKMEDIATE   ARITHMETIC. 


30.  Reduce  .6349£-f  hhd.  to  a  compound  number. 

Ans.  40  galls. 

31.  Reduce  .000124^^  Tun  to  a  compound  num- 
ber. Ans.  1  gill. 

32.  Reduce  .0198-^g-  Tun  to  a  compound  num- 
ber. Ans.  2  qt. 

33.  Reduce  20.790625  £  to  a  compound  number. 

Ans.  20  £  15s.  9d.  3  far. 

34.  Reduce  1.4107l4ftoa  compound  number. 

Ans.  1  hhd.  25  gal.  3  qt.  1  pt. 

35.  Reduce  1.184  Ton  to  a  compound  number. 

Ans.  23  cwt.  2  qr.  18  Ibs. 

36.  Reduce  .499  Ton  to  a  compound  number. 

Ans.  9  cwt.  3  qr.  23  Ibs. 

37.  Reduce  14.22708^  £  to  a  compound  number. 

Ans.  14£.  4s.  6d.  2  far. 

38.  Reduce  .000806640625   Ton  to  a  compound 
number.  Ans.  I  Ib.  9  oz.  13  dwt. 

39.  Reduce  .999  Ton  to  a  compound  number. 

Ans.  19  cwt.  3  qr.  23  Ib. 

40.  Reduce  .2996-^  hhd.  to  a  compound  number. 

Ans.  18  gall.  3  qts.  1  pt. 

41.  Reduce  l.OOOff-f  yr.  to  a  compound  number. 

Ans.  365  da.  5  h.  48  m. 

42.  Reduce  .7617^  mile  to  a  compound  number. 

Ans.  4  A.  3  R.  20  rods. 

43.  Reduce  2.31  cwt.  to  a  compound  number. 

Ans.  2  cwt.  1  qr.  6  Ib. 

44.  Reduce  .3036|f  deg.  to  a  compound  number. 

Ans.  21  miles. 


REDUCTION   OF   DENOMINATE    DECIMALS.          325 

45.  Reduce  .0309  R.  to  a  compound  number. 

Arts.  I  sq.  rod  57  sq.  ft. 

46.  Reduce  12.619  Guineas  to  a  compound  num- 
ber. Ans.  13  £.  5s. 

47.  Reduce  .7375  £.  to  a  compound  number. 

Ans.  14s.  9d. 

48.  Reduce  .8925  Ton  to  a  compound  number. 

Ans.  17  cwt.  3  qrs.  10  Ib. 

49.  Reduce  10.720|-  £.  to  a  compound  number. 

Ans.  10  £.  14s.  Od.  2  far. 

50.  Reduce  1.5125  mile  to  a  compound  number. 

Ans.  1  m.  4  fur.  4  r. 

51.  Reduce  5.875  Acre  to  a  compound  number. 

Ans.  5  A.  3  R.  20  rods. 

52.  Reduce  .4405  Ton  to  a  compound  number. 

Ans.  8  cwt.  3  qr.  6  Ibs. 

53.  Reduce  .81  cwt.  to  a  compound  number. 

Ans.  3  qr.  6  Ibs. 

54.  Reduce  .67586f-f-  Ib.  to  a  compound  number. 

Ans.  8  oz.  2  dwt.  5  grs. 

55.  Reduce  2.1875  yds.  to  a  compound  number. 

Ans.  2  yds.  3  na. 

56.  Reduce  .2541 6f  deg.  to  a  compound  number. 

Ans.  2°  15'  15". 

57.  Reduce  20.41 6f  Ib.  to  a  compound  number. 

Ans.  20  Ib.  10  dwt. 

58.  Reduce  VOO^2^-  yr.  to  a  compound  number. 

Ans.  255  da.  16  h.  2  m.  10  s. 

59.  Reduce  94.383|-  £  to  a  compound  number. 

Ans.  94  £  7s.  8d. 
28 


326  INTERMEDIATE  ARITHMETIC. 

60.  Reduce  2.63|-  £  to  a  compound  number. 

Ans.  2£  12s.  8d. 

61.  Reduce  2.46875  bu.  to  a  compound  number. 

Ans.  2  bu.  1  pk.  7  qts. 

62.  Reduce  10.284  Ton  to  a  compound  number. 

Ans.  10  Ton  5  cwt.  2  qr.  18  Ib. 

63.  Reduce  84.30401 6|  deg.  to  a  compound  num- 
ber. Ans.  84°  18'  15". 

64.  Reduce  .2341 7T8^  circ.  to  a  compound  num- 
ber. Ans.  84°  18'  15". 

65.  Reduce  .0119|-|i  circ.  to  a  compound  number. 

Ans.  4°  18'  15". 

66.  Reduce  .05  £  to  a  compound  number. 

Ans.  I  shilling. 


ADDITION  OF  DENOMINATE  NUMBERS. 
LESSOX  XXVII. 

EULE    I. 

341.  I.  Write  the  quantities  so  that  the  figures 
of  the  same  denomination  icill  be  found  in  the  same 
column. 

II.  Add  the  columns  separately,  beginning  with 
the  right-hand  column. 

III.  After  having  found  the  sum  of  the  first  col- 
umn,  see  how  many  units  of  the  next  column  are  con- 
tained  in  the  sum  of  the  first.      Write  the  remainder 
under  the  first  column  and  carry  the  quotient  to  the 
second  column. 


ADDITION   OF   DENOMINATE   NUMBEES.  327 

IV.  Add  the  figures  in  the  second  column.     See 
how  many  units  of  the  next  column  are  contained  in 
the  sum  of  the  second  column.      Write  the  remainder 
under  the  second  column,  and  carry  the  quotient  to 
the  next. 

V.  Continue  in  this  way  until  the  last  column  is 
reached.    Add  the  figures  in  the  last  column,  and 
write  the  sum  under  the  last  column. 


NOTE. — If  fractional  parts  of  dcnoirinatc  numbers  arc  used, 
reduce  the  fractions  to  compound  numbers,  and  add  as  before. 

342.  NOTE  TO  THE  PUPIL. — In  adding  sums  in  Long  Measure 
and  also  in  Square  Measure  the  attention  of  the  pupil  is  necessary 
to  the  following  remarks : 

"Inches  and  feet  being  of  an  even  denomination,  no  difficulty 
is  experienced  in  dividing  the  sum  of  the  columns ;  but  when  the 
column  of  yards  is  added,  the  sum  has  to  be  divided  by  a  mixed 
number,  that  is,  by  a  whole  number  and  a  fraction.  The  remainder 
in  this  case  is  different  from  all  others,  because  the  remainder,  if 
any,  is  not  yards,  but  a  fraction  of  a  rod.  This  fraction  of  a  rod 
must  be  multiplied  by  5^  (if  Long  Measure),  or  30^  (if  Square 
Measure),  to  find  the  number  of  yards.  For,  since  the  remainder 
is  a  fraction  of  a  rod,  to  reduce  that  fraction  to  a  lower  denomina- 
tion, it  is  necessary  to  multiply  it  first  by  54-  (for  Long  Measure), 
or  by  SOA  (for  Square  Measure),  to  find  the  number  of  yards.  In 
many  examples  the  answer  results  in  yards  and  a  fraction  of  a  yard. 
This  fraction  of  a  yard  must  be  reduced  to  feet,  by  multiplying  it 
by  3  (for  Long  Measure),  or  by  9  (for  Square  Measure),  and  the 
number  thus  found  must  be  added  to  the  number  found  under  the 
column  of  feet.  If  a  fraction  should  occur  with  the  feet,  reduce 
it  to  inches  by  multiplying  it  by  12  (for  Long  Measure),  or  by 
144  (for  Square  Measure),  the  result  to  be  added  to  the  number 
found  under  the  column  of  inches.  Having  completed  the  several 


328  INTERMEDIATE  ARITHMETIC. 

operations  as  above  indicated,  examine  the  number  under  the 
column  of  inches  and  see  if  there  are  too  many  inches.  If  so 
reduce  them  to  feet,  write  the  remainder  in  place  of  the  figures 
under  the  column  of  feet,  and  add  the  quotient  to  the  figures  under 
the  column  of  feet.  See  if  there  are  more  (feet)  than  would  make 
a  yard.  If  so,  reduce  the  feet  to  yards,  write  the  remainder,  if 
any,  in  place  of  the  number  under  the  column  of  feet,  and  add  the 
quotient  to  the  number  under  the  column  of  yards." 

In  adding  the  fractional  parts  of  Denominate 
numbers,  as  \  mile  and  \  of  a  mile,  add  the  fractions, 
thus,  J+i-=yV  mi^e5  if  they  are  of  the  same  value  as 
above,  and  then  find  the  value  of  the  fraction  thus 
found. 

In  the  above  example  \  of  a  mile  and  ^  of  a  mile 
are  equal  to  T7^-  of  a  mile,  which,  being  reduced, 
gives  4  fur.  26  r.  3  yds.  2  ft.  0  in. 

The  same  remarks  apply  to  decimal  fractions  of 
Denominate  Numbers. 

If  the  fractio-ns  are  of  different  denominate  value, 
they  may  be  reduced  to  the  same  value  or  denomin- 
ation, thus  -J-  pipe  and  \  Tun. 

J  of  a  pipe  is  equal  to  \  of  a  Tun,  because  it  re- 
quires 2  pipes  to  make  a  tun,  and  \  divided  by  2  is 
the  same  as  \  X  \  or  \  of  a  Tun. 

We  now  have  -J-  Tun  and  J-  of  a  Tun  to  be  added 

2-4-3 
together  =  or— -£<-=:•&  Tun,  which,  when  reduced, 

would  give  I  hhd.  and  42  galls. 
Or: 

Reduce  the  Tuns  in  the  above  example  to  pipes, 
thus  J-  pipe  and  J  Tun  would  be  the  same  as  }  pipe 


ADDITION    OF   DENOMINATE   NUMBERS.  329 

and  £  of  a  pipe.  To  reduce  Tuns  to  pipes,  we  must 
multiply  the  fraction  by  2,  because  2  pipes  make  a 
tun.  By  multiplying  £  by  2,  we  get  f-  or  £.  Then 
adding  the  fractions  •£— hi^f  of  a  pipe,  which  re- 
duced, gives  1  hhd.  and  42  galls. 


ADDITION  OF   DENOMINATE   NUMBERS. 

(1)  (2) 

£.    s.    d.  far.  Ton.  cwt.  qrs.  Ibs.  oz.  dram. 

1522  103581 

351  212146 

4683  1      18       3     16     10     10 

5   15     42  510     23       71 


(3)  W 

Ib.  oz.  dwt.  grs.  m.  fur.  r.  yds.  ft.  in. 

1   3  12  21  133214 

9  15  22  138416 

8  10  10  23  7  20   1   2   9 


10  11   19  18       3.   5  32   3   1 


(5)  (6) 

£  s.   d.  far.  sq.  m.  A.  R.  r.  yds.  ft.  in. 

3  14  11  3     1  604  3  19  21  8  100 

1   10   3  3       168  2  24  20  6  190 

15   10  3        60  3  32  10  5  36 


6121     2  194  1  36  23  1 

28* 


330 


INTERMEDIATE   ARITHMETIC. 


Ill 


(1)  (8) 

.  fur.  r.  yds.  ft.  in.  Ton.  cwt.  qr.  Ib.  oz.  dr. 

22  20  425  .2  12  2   6  14  5 

1  4  25  5  2  G  1   8  3  12  12  14 

5  35  4  1  8  2  20  12  2 


45   2407 


4   2  0  15   7 


(9)  (10) 

Ib.  oz.  dwt.  gr.  sq.  m.  A.  R.  r.  yds.  ft.  in. 

3  2   8  20  1   64  2  30  20  8  120 

3  1   4  20  2  136  3  15   6  7  20 

8  10  12  583  2  30  29  6  83 


7  0 


4  145  0  36  27  2 


43 


(11) 


(12) 


Ibs.  oz.  dwt.  grs.  sq.  m.  A.  R.  r.  y.  ft.  in. 

4  3   2   20  1   64  3  31  21  7  120 

1  4   8   21  590  2  39  28  6  132 

264  3  28  15  5  72 

1   13 


ADDITION   OF   COMPOUND   NUMBERS. 
(1) 

£.    s.     d.  far. 
1     3     10     2 


2     3 
4     6 


(2) 

m.  fur.     r.      y.     ft.     in. 
123524 


3        8 

7     20 


G 


7  14       02 


3       4     33 


ADDITION   OF   COMPOUND   NUMBERS.  331 


(3) 

(4) 

Ton.  cwt.  qrs.  Ibs.  oz 

.  dr. 

£.  s.   d.  far. 

12352 

1 

3   4  11  3 

21314 

6 

1  16   3  3 

1   18   3  16  12 

10 

15   8  2 

5   3   1  23   3 

1 

5  17   0  0 

(*) 

(6) 

m.  A.  R.  r.  y.  ft. 

in. 

m.  fur.  r.  y.  ft.  in. 

1  604  3  29  21  7 

100 

2  3  20  4  2  5 

62  2  24  20  6 

190 

1  4  25  5  2  6 

60  3  14  16  5 

118 

5  30  4  1  11 

2   93  1  28   90 

84 

4  5  37  4  0  10 

en 

(8) 

Ton.  cwt.  qr.  Ibs.  oz. 

dr. 

Ibs.  oz.  dwt.  grs. 

2  12  2   6   4 

2 

3   2   8  20 

1   8  3  12  12 

14 

3  10  14  22 

8  2  20  14 

2 

8  10   2 

4  10  0  13  15 

2 

7   9  18  20 

(9) 

(10) 

Ib.  oz.  dwt.  grs.  m. 

A. 

R.  r.  y.  ft.  in. 

1   3  12  20    1 

64 

2  30  20  8  121 

2   9  15  21    2 

100 

3  25  16  7  120 

8  10  19  22 

583 

2  39  29  6   283 

13       0       8     15         4     109     1     16       7     2       20 


332 


INTERMEDIATE   ARITHMETIC. 


(11)  (12) 

Tun.  pi.  hd.  gal.  qt.  pt.  gi.    Tun  pi.  hd.  gall.  qt.  pt.  gi. 


1  1  1  60  2  1  2 

1  1  1  20  3  1  2 

1  1  1  30  2  1  2 

1  1  1  20  2  1  2 


1      1      1  60     2     0     2 

1      0      1  30     0     1      0 

1      0     0  40     2     2     2 

110  0222 


6     0 

0       6 
(13) 

] 

.     0     2 

7 

10700 
(14) 

0 

hhd.  bbl.  gall 

.  qts.  pts. 

qr. 

bu, 

P- 

qt. 

pt. 

1 

1 

24 

2       1 

1 

4 

2 

4 

1 

1 

2 

30 

1        2 

1 

5 

1 

5 

1 

1 

1 

30 

2       1 

6 

2 

6 

1 

1 

1 

20 

1        1 

1 

1 

2 

1 

8 

1 

34 

0       1 

4 

2 

0 

3 

0 

(15) 

(16) 

qr. 

bu. 

p.  qt. 

pt.     yr. 

mo. 

da.  hr. 

sec. 

mi. 

1 

4 

2 

4 

1         1 

2 

3 

2 

15 

20 

5 

3 

5 

1 

10 

21 

20 

40 

30 

4 

6 

2 

20 

18 

40 

40 

2 

3 

3 

0 

1          2 

1 

15 

17 

36 

30 

(17) 

yr.    da.     hrs.  mi.  sec. 

1  200       3  2  1 
300     29  58  59 

20     30  30  22 

1  30  40 

2  157     17  2  2 


(18) 
circ.  S. 

1  6     15     30  30 
3        7     15  15 

20     40  50 

2  10     13     26  35 


ADDITION    OF   COMPOUND   NUMEEKS.  333 


(19) 
circ.  S.     ° 
1     6     20     30 

ft). 

40           1 

(20) 
I      3      3 
6       4       1 

grs. 
10 

9     30     18 

50 

981 

15 

50     30 
12 

40 

6       2 

10 

22            2 

532 

15 

2     6     11     32 

32 

(21) 

(22) 

1972 

grs.       yd, 
18         1 

qrs.    na. 
2         3 

in. 
2 

8       6        1 

16 

3         2 

1 

5        2 
1 

18 

1 

2 

20         2 

3         0 

* 

2750 

12 

(23) 

yd. 

1 

qrs.      na. 
2         1 

in. 
1 

3         3 

2 

2 

1 

24.  Add  ^  mile  and  %  mile. 

^.ws.  4  fur.  26  r.  3  yds.  2  ft.  0  in. 

25.  Add  |  rod  and  -J-  rod.     ,4ws.  3  yd.  2  ft.  6|  in. 

26.  Add  f  hhd.  and  |  hhd. 

Ans.  57  galls.  1  quart  1  pint. 

27.  Add  ^  pipe  and  £  Tun.    Ans.  1  hhd.  42  galls. 


334 


INTERMEDIATE   AEITHMETIC. 


28.  Add  |  A.  and  f  R. 

Ans.  3  R.  39  r.  30  yds.  2  ft.  36  in. 

29.  Add  -J  Ton  and  -J-  cwt. 

Ans.  3  cwt.  1  qr.  19  Ibs.  7  oz.  If  drs. 

30.  Add  f  Ton  and  f  cwt. 

Ans.  17  cwt.  1  qr.  8  Ibs.  5  oz.  5J  drs. 

31.  Add  J  qr.  and  £  Ib. 

-4?zs.  12  Ibs.  13  oz.  5J  drs. 

32.  Add  i£  and  ^s.  Ans.  2s.  7d. 

33.  Add  f£  and  |d.  .4ws.  16s.  8d.  2f  far. 

34.  Add  f  £  and  -J-  far.  ^is.  13s.  4d.  J-  far. 

35.  Add  J  sign  and  f  degree. 

Ans.  15  degrees  40  mi. 

36.  Add  -J  sq.  yd.  and  f  sq.  ft.     Ans.  8  ft.  108  in. 

37.  Add  -J-  bu.  and  -j-  pk.        Ans.  7  qts.  1T7¥  pints. 

38.  Add  ^  jr.  and  ^  day.  ^TIS.  121  days  20  hours. 

39.  Add  |  century  and  f  year. 

Ans.  11  yr.  9  mo.  10  da. 

40.  Add  .025£.  and  .024  sh.       Ans.  6d.  1.152  far. 

41.  Add  .35£  and  .25  sh.  Ans.  12s.  3d. 

42.  Add  .025  Ton  and  .02  Ib. 

Ans.  2  qr.  0  Ibs.  0  oz.  5.12  drams. 

43.  Add  .05  cwt.  and  .033|  qr. 

Ans.  5  Ibs.  13  oz.  51  drams. 

44.  Add  .0352  mile  and  .024  rod. 

Ans.  11  rods  1  yd.  1  foot  9.024  in. 

45.  Add  .02  sq.  m.  and  .55  R. 

Ans.  12  A.  3  R.  30  rods. 

46.  Add  .186  sq.  yd.  and  .5  sq.  ft. 

Ans.  2  sq.  ft.  25.056  in. 


SUBTRACTION    OF   COMPOUND   NUMUEES.  335 

47.  Add  .025  day  and  .025  hr. 

Ans.  37  mi.  30  sec. 

48.  Add  .0252  yr.  and  .0252  da. 

Ans.  9  da.  5  hrs.  21  m.  24.48  sec. 

49.  Add  .05°  and  .005°.  Ans.  3  mi.  18  sec. 

50.  Add  .005°  and  .025°.  Ans.  1  mi.  48  sec. 

51.  Add  .025  hhd.  and  .03  gall. 

Ans.  1  gall.  2  qrts.  0  pints  3.36  gills. 

52.  Add  .066  Tun  and  .024  hhd. 

Ans.  18  galls.  0  qts.  1  pint.  0.628  gill. 

53.  Add  .4405  Ton  and  .01  cwt. 

Ans.  8  cwt.  6  qrs.  12  Ibs. 

54.  Add  .875  A.  and  .0309  R. 

Ans.  3  R.  21  r.  7  y.  1  ft.  36.144. 

55.  Add  .999  Ton  and  .499  Ton. 

Ans.  1  Ton  9  cwt.  3  qrs.  21  Ibs. 

56.  Add  .284  Ton  and  .8925  Ton. 

Ans.  1  Ton  3  cwt.  2  qr.  3  Ibs 


SUBTRACTION  OF  COMPOUND  NUMBERS. 
LESSON  XXIX. 

342.  Subtraction  of  compound  numbers  is  the 
process  of  finding  the  difference  between  two  denom- 
inate numbers  of  the  same  kind. 


336 


INTERMEDIATE   ARITHMETIC. 


EXAMPLE   IN   ENGLISH   MONET. 

£          s.         d.      far. 
From        78         11         5         2 
Take         42         14         3         3 

35         17         1         3 

Having  placed  the  less  number  under  the  greater, 
farthings  under  farthings,  etc.,  we  begin  and  sub- 
tract, from  farthings.  3  farthings  from  2  farthings 
we  cannot ;  borrow  1  penny,  which  is  equal  to  4 
farthings,  and  2  farthings  we  have  make  6  farthings. 
Now  3  far.  from  6  far.  leaves  3  far.  Having  bor- 
rowed 1  penny  from  the  column  of  pence,  we  have  to 
subtract  1  from  the  5  in  the  minuend  or  add  1  to  the 
3  in  the  subtrahend.  It  is  more  convenient  to  add 
1  to  the  figure  in  the  subtrahend.  Say,  3  and  1  I 
borrowed  are  4  pence.  4  from  5  leaves  1.  Write 
the  1  under  the  column  of  pence.  As  we  did  not 
require  to  borrow  any  thing,  we  have  nothing  to  add. 
Say,  14  shillings  from  11  shillings,  we  cannot;  bor- 
row 1  pound  sterling,  which  is  equal  to  20  shillings 
and  11  we  have  make  31  shillings.  14  from  31 
leaves  17,  which  we  write  in  the  column  of  shillings, 
and  carry  1  to  the  next  higher  number  in  the  sub- 
trahend. 42  and  1  are  43.  43£  from  78£  leaves 
35£.  The  answer  as  found  is  35£  17s.  Id.  3  far. 

Or: 

Reduce  the  £,  etc.  of  the  minuend  to  farthings, 


and  reduce  the  subtrahend  to  farthings.   Subtract  one 


SUBTRACTION    OF   COMPOUND   NUMBERS.  337 

from  the  other.     Reduce  the  farthings  thus  found  to 
the  higher  denominations  for  the  answer. 

RULE  I. 

Write  the  less  denominate  number  wider  the  great- 
er, so  that  the  units  of  the,  same  denomination  may 
stand  under  each  other. 

Subtract  as  in  simple  subtraction. 

If  any  number  in  the  subtrahend  is  greater  than 
the  corresponding  number  in  the  minuend,  add  to  the 
number  in  the  minuend  as  many  units  as  makes  one 
of  the  next  higher  denomination,  and  carry  1  to  the 
next  higher  figure  in  the  subtrahend  after  you  subtract. 

Proceed  in  this  manner  to  the  end  of  the  sum. 

RULE  II. 

Heduce  the  numbers  to  the  lowest  denomination. 
Subtract  one  from  the  other,  and  reduce  the  remainder 
to  the  higher  denominations  for  the  answer. 

343.  The  pupil  should  be  careful  and  make  no 
mistakes  in  Long  Measure,  Square  Measure,  or  any 
other  measure  having  fractions. 

344.  To  subtract  denominate  fractions,  reduce 
them  to  the   several   denominations,   and   subtract 
according  to  Rule  I.  or  II. 

345.  The  same  remark  applies  to  denominate 
decimal  fractions. 

29 


338 


INTEKMEDIATE  AEITHMETIC. 


EXAMPLES. 

(1.) 

(2.) 

£   s.  d.  far. 
From  78  10   4   1 
Take  41  11   5   2 

£   s.   d.  far. 
65   6   9   1 
13   7  10   2 

36  18  10   3 

51  18  10   3 

(3.) 

(4.) 

Ibs.  oz.  dwt.  grs.      Ton 
15   3  10  12        17 

cwt.  qr.  Ibs.  oz.  drs. 
16   2   4   0   12 

9  11  12  14         7 

17   3   5   1   13 

5   3  17  22         9 

18   2  23  14   15 

(5.) 

(6.) 

Ton  cwt.  qr.  Ibs.  oz.  drs. 
10   1   0   1   1  12 

£  s.   d.  far. 
8  10   3   2 

9  18   2   4   3  14 

4  11   5   3 

2   1  21  13  14 

3  18   9   3 

(7.) 

(8.) 

Ibs.  oz.  dwt.  grs. 
2   4  10  20 

Ibs.  oz.  dwt.  grs. 
3   5  11   21 

1   5  12  21 

2   6  12   23 

10  17  23 

10  18   22 

SUBTRACTION   OF   COMPOUND   NUMBERS.  339 

(9.)  (10.) 

Ton  cwt.  qr.  Ibs.  oz.  drs.  Ton  cwt.  qr.  Ibs.  oz.    drs. 

2        32     20     8  4  3  21      18  50 

143      22     9  5  4  32     20  6        Y 


18     2     22  14     15  18     2     22     14     15 

(11.)  (12.) 

ft)   i   3   3  grs.  ft>   3   3   3  grs. 

26425  32514 

17536  13625 


10   6   1  19  1  10   6   1  19 

(13.)  (14.) 

m.  fur.  r.  y.  ft.  in.  m.  fur.  r.  y.  ft.  in. 

24  10  214  425326 

15  11  325  336437 


6     38     405  6     38     405 

(15.) 

L.       m.      fur.       r.  yds.     ft.      in. 

1         3         4         20  2         1         8 

1          2         5          21  3                    9 


6 

38 

4 

0 

5 

(16.) 

m.     A. 

R. 

r. 

y- 

ft. 

in. 

L          0 

2 

30 

10 

4 

72 

3 

40 

20 

5 

100 

639          2         29         20 


340 


INTERMEDIATE    ARITHMETIC. 


(17.) 


sq.  m. 

A. 

R. 

r. 

y. 

ft. 

in. 

1 

0 

1 

25 

10 

2 

80 

l 

30 

5 

3 

100 

639 

3 

35 

4 

7 

124 

(18.) 

sq.  m. 

A. 

R. 

r. 

y- 

ft. 

in. 

4 

0 

1 

10 

4 

5 

8 

2 

0 

1 

11 

5 

6 

80 

1 

639 

3 

38 

29 

0 

108 

(19.) 

Tun    pi. 

hhd. 

galls. 

qts. 

pt. 

gills. 

1 

1 

1 

16 

2 

1 

2 

1 

0 

30 

3 

0 

3 

(20.) 

galls,  qrts.  pt.  gills. 
16       2       1       2 
5303 


10 


0 


(22.) 

bu.     pk.    qt.  pt. 

12       2       5  1 

10       3       6  1 

1       2       1  0 


0          48          3          0          2 

(21.) 

qr.    bu.  pks.  qrts.  pt. 
14241 
5351 

6270 

(23.) 

yr.  mo.  da.  hrs.  mi.  sec. 
4'  5  4  10  20  20 
3  6  5  11  30  35 

10  28  22  49  45 


SUBTRACTION    OF   COMPOUND  NUMBERS.  341 


(24.)  (25.) 

yrs.  mo.  da.  hrs.  mi.  sec.  w.     da.   hr.    mi. 

4       2     12       5  10  10  4235 

2       3     15       6  11  12  2346 


1      10     26     22     58     58  15     22     59 


(26.)  (27.) 

circ.    S.  deg.     '  "  S.  cleg.    '  " 

4       3     10     20  20  4  3     14  21 

2       4     11     21  21  2  4     16  22 


1     10     28     58     59  1     28     57     59 


(28.)  (29.) 

£      s.     d.      far.  circ.    S.   deg.    '       " 

1342  21412 

12       5       3  2513 


10     10       3  1     10     28     59     59- 


(30.) 

m.     fur.      r.       yds.    ft.       in. 

121216 

40307 

60425 

29* 


342  INTERMEDIATE  ARITHMETIC. 


LESSON  XXX. 

31.  From  f  mile  subtract  -J-  mile.  Ans.  4  furlongs. 

32.  From  f  mile  subtract  ^  mile. 

Ans.  2  fur.  26  r.  3  yds.  2  ft. 

33.  From  f  £  subtract  1  £. 

Ans.  6  shillings  8  pence. 

34.  From  4-  £  subtract  £  sh. 

^4ws.  9  shillings  9  pence. 

35.  From  |  £  subtract  |d.        -4ws.  2s.  5d.  2|  far. 

36.  From  \  £  subtract  ^d.        Ans.  4s.  lid.  2  far. 

37.  From  ^  £  subtract  ^-  sh.  Ans.  6s.  6d. 

38.  From  ^  Ib.  subtract  -|  oz.        Ans.  7  oz.  8  drs. 

39.  From  ^  Ib.  subtract  -J-  Ib.     <4ws.  2  oz.  10J  drs. 

40.  From  £  Ib.  subtract  |-  Ib.         -4»s.  2  oz.  0  drs. 

41.  From  ^  Ib.-subtract  ^  oz.  Ans.  2  oz  13-j-f  drs. 

42.  From  -J-  Ib.  subtract  -J-  oz.    Ans.  2  oz.  7T\  drs. 

43.  From  \  Ib.  subtract  ^  oz.      .4ws.  2  oz.  2^  drs. 

44.  From  J  ton  subtract  ^  cwt.  «4«a.  4  cwt.  2  qrs. 

45.  From  \  cwt.  subtract  \  Ib. 

,4ws.  1  qr.  24  Ibs.  8  oz. 

46.  From  \  qr.  subtract  \  Ib. 

Ans.  5  Ib.  14  oz.  lOf  drs. 

47.  From  ^  Ib.  subtract  ^  oz.      Ans.  5  oz.  3^  drs. 

48.  From  ^  oz.  subtract  £  oz.  As.  4  drams. 

49.  From  1  league  subtract  ^  mile. 

^ws.  7  furlongs. 

50.  From  J  mile  subtract  ^  yard.' 

As.  1  fur.  39  rods  5  yds. 


SUBTRACTION   OP  COMPOUND  NUMBERS.  343 

51.  From  -J-  fur.  subtract  J  rod. 

Ans.  G  rods  1  yd.  2  ft.  6  in. 

52.  From  -J-  rod  subtract  £  yard.  Ans.  GJ-. 

53.  From  £  sq.  mile  subtract  £  sq.  mile. 

^ws.  106  A.  2  R.  26  r.  20  yds.  1  ft.  72  in. 

54.  From  -J  rood  subtract  £  rod. 

^ws.  7  rods  15  yds.  1  ft.  18  in. 

55.  From  £  sq.  yd.  subtract  J  foot, 

-4«s.  4  feet  24  inches. 

56.  From  -J-  sq.  foot  subtract  £  inch.   Ans.  47 f-  in. 

57.  From  £  tun  subtract  £  hhd. 

Ans.  10  galls.  2  qts. 

58.  From  J  gall  subtract  -J-  gill. 

^M5.  1  pt.  3f  gills. 

59.  From  J  barrel  subtract  £  barrel. 

Ans.  7  galls.  3  qts.  1  pint. 

60.  From  J  bushel  subtract  -}-  peck. 

^4ws.  1  pk.  1  qr.  |  pint. 

61.  From  £  bushel  subtract  ^  bushel. 

-<4ws.  3  qrts.  f  pint. 

62.  From  £  year  subtract  £  day.    Ans.  182  days. 

63.  From  J  day  subtract  £  hour. 

Ans.  5  hrs.  30  mi. 


LESSON"  XXXI. 
64.  From  .05  £  take  .005  £. 


.  10  pence  3.2  far. 
65.  From  .25  mile  take  .025  mile. 

Ans.  1  furlong  32  rods. 


344  INTERMEDIATE   ARITHMETIC. 

66.  From  .75  £  take  .075  £. 

Ans.  13  sh.  6  pence. 
C7.  From  .05  £  take  .05  sh. 

Ans.  11  pence  1.6  far. 

68.  From  .525  £  take  .52  sh. 

^l?is.  9  sh.  11  pence  3.04  far. 

69.  From  .05  Ib.  take  .05  oz.  Jlrcs.  12  drams. 

70.  From  .025  Ib.  take  .025  oz.         Ans.  6  drams. 

71.  From  .02  Ib.  take  002  Ib.      Ans.  4.608  drams. 

72.  From  .0025  Ib.  take  00025  Ib. 

Ans.  0.576  drams. 

73.  From  .06  Ib  take  .OOB  Ib.  Ans.  14.592  drams. 

74.  From  .02  ton  take  .02  qr. 

Ans.  1  qr.  14  Ibs.  8  oz. 

75.  From  .05  cwt.  take  .05  Ib. 

Ans.  4  Ibs.  15  oz.  3.2  drams. 

76.  From  .5  qr.  take  .05  Ib. 

Ans.  12  Ibs.  7  oz.  3.2  drams. 

77.  From  .5  Ib.  take  .05  drams. 

Ans.  7  oz.  15.95  drams. 

78.  From  .5  oz.  take  .05  dram.     Ans.  7.95  drams. 

79.  From  .025  mile  take  .025  fur.        Ans.  7  rods. 

80.  From  .025  fur.  take  .25  yard. 

Ans.  5  yds.  0  ft.  9  in. 

81.  From  .025  rod  take  .025  foot.       Ans.  4.65  in. 

82.  From  .25  yard  take  .05  foot.  Ans.  8.4  in. 

83.  From  .033-1  Sq.  m<  take  .024  R. 

Ans.  21  A.  1  R.  13  r.  10  yd.  0  ft,  31.104  in. 

84.  From  .024  R.  take  .033^  yd. 

Ans.  3.552  inches. 


SUBTRACTION   OF   COMPOUND   NUMBERS.  345 

85.  From  .03-J-  yd.  sq.  take  033£  ft. 

Ans.  38.4  inches. 

86.  From  .080  tun  take  .025  gall. 

Ans.  21  galls.  1  qt.  1  pt.  3.704  gill. 

87.  From  .025  gall,  take  .05  barrel.    Ans.  0.4  gill. 

88.  From  .05  barrel  take  .052  gall. 

Ans.  1  gall.  1  qt.  1  pt.  4  gill  .730. 

89.  From  .02  league  take  .02  mile. 

Ans.  12  rods  4  yds.  1  ft.  2.4  inches. 

90.  From  .02  mile  take  .02  fur. 

Ans.  5  rods  3  yds.  0  ft.  10.8  inches. 

91.  From  .02  fur.  take  .02  rod. 

Ans.  4  yd.  Oft.  10.44  in. 

92.  From  .02  rod  take  .02  yard.  Ans.  3.24  inches. 

93.  From  .02  yard  take  .02  foot.     Ans.  0.48  inch. 

94.  From  .02  foot  take  .02  inch.     Ans.  0.22  inch. 

95.  From  .02  in.  take  .002  inch.      Ans.  .018  inch. 
90.  From  .5864  gall,  take  .26  gall. 

Ans.  1  qt.  0  pi.  2.4448  gill. 

LESSON"  XXXII. 
347.  To  find  the  difference  of  dates. 

1.  What  is  the  difference  of  time  between  Nov. 
13th,  1864  and  Aug.  llth,  1869. 

RULE. 

Place  the  earlier  date  under  the  later. 
Consider  30  days  as  one  month  and  360  days  as 
one  year. 

Subtract  one  from  the  other  by  the  rule. 


346 


INTERMEDIATE  ARITHMETIC. 


OPEEATION. 

year. 
1869 

month. 

7 

day. 
11 

1864 

10 

13 

Write  the  year  of  the  later  date,  1869.  Begin 
and  count  the  number  of  months  from  January  to 
August.  Thus,  January,  1 ;  February,  2 ;  March, 
3  ;  April,  4  ;  May,  5  ;  June,  6  ;  July,  7.  We  find  7 
months  to  have  elapsed.  Write  down  the  figure  7  a 
little  to  the  right  of  the  year,  as  above,  for  months. 
Write  the  figure  13  a  little  to  the  right  of  the  figure 
7,  as  above,  for  the  days.  Find  the  time  for  the 
minuend,  and  subtract  one  from  the 'other. 


EXAMPLES  FOE  PEACTICE. 

2.  A  note   dated  March  3d,  1860,  became  due 
Oct.  3d,  1868.     How  long  was  it  on  interest? 

3.  Washington  was  born  Feb.   22d,   1732,   and 
died  Dec.  14th,   1799.    How  old  was  he  when  he 
died? 

4.  America  was  discovered  by  Columbus,  Oct. 
12th,  1492.    What  time  elapsed  from  that  time  to 
the  birth  of  Geo.  Washington  ?    To  his  death  ? 

5.  The  Declaration  of  Independence  was  signed 
July  4th,  1776.     What  time  elapsed  between  that 
event  and  the  death  of  Washington  ? 

6.  How  old  was  Washington  when  the  colonies 
declared  themselves  free  and  independent  ? 


MULTIPLICATION   OF   COMPOUND  NUMBERS.        347 


MULTIPLICATION  OF  COMPOUFD  NUMBERS. 

LESSON  XXXIII. 

348.  Multiplication  of  Compound  Numbers  is 
the  process  of  taking  a  given  compound  number  any- 
given  number  of  times. 

There  are  two  methods.  The  pupil  may  choose 
that  which  he  understands  best. 

As  one  method  is  a  proof  of  the  ether,  it  would 
be  well  to  make  each  and  every  sum  by  the  two 
rules. 

EXAMPLE. 

Multiply  2£  4s.  3d.  2  far.  by  8. 

OPERATION. 


£ 

s. 

d. 

far. 

2 

4 

3 

2 

8 

17  14  4  0 

Multiply  as  in  multiplication  of  abstract  numbers. 
8  times  2  far.  are  16  far.  4  far.  make  a  penny.  4 
into  16,  4  times  and  0  remaining.  Write  the  0 
under  the  farthings,  and  carry  the  4  to  be  added  to 
the  next  result.  8  times  3d.  are  24d.  and  4  are  28d. 
12  pence  make  a  shilling.  12  into  28,  2  times  and 
4d.  over.  Write  the  4d,  under  the  pence,  and  carry 


348  INTERMEDIATE   ARITHMETIC. 

the  2  to  be  added  to  the  next  result.  8  times  4s.  are 
32s.  and  2  are  34s.  20  shillings  make  1  £.  20  into 
34  goes  1  time  and  14  shillings  over.  Write  the  14 
under  the  shillings,  and  carry  the  1  to  be  added  to 
the  next  result.  8  times  2£  are  16£,  and  1  are  17£, 
which  is  written  under  the  column  of  £.  The  an- 
swer is  17£  12s.  4d.  0  far. 

349.  Or,  reduce  the  compound  number  to  its 
lowest  denomination,  multiply,  and  reduce  the  result 
to  the  higher  denominations. 

In  the  above  example,  2£  4s.  3d.  2  far.  are  equal 
to  2126  farthings,  which  multiplied  by  8  gives  17008 
farthings.  Reducing  17008  farthings,  we  get  17£ 
14s.  4d.  0  far.  for  the  answer. 

RULE. 

I.  Multiply  each  denomination  of  the  compound 
number  as  in  multiplication  of  abstract  numbers. 

II.  Find  how  many  of  the  next  higher  denomina- 
tion is  contained  in  each  result,  and  add  it  to  the 
result  found  for  the  next  higher  denomination. 

Or: 

Reduce  the  compound  number  to  its  lowest  denom- 
ination, multiply  the  result  thus  found  by  the  multi- 
plier, after  which  reduce  the  result  to  the  several  higher 
denominations. 


MULTIPLICATION    OF   COMPOUND   NUMBERS.       349 


EXAMPLES  FOR   PRACTICE. 

(1.)  (2.) 

£    s.    d.  £     s.  (1. 

248  9     11  7 

3  4 


6  14     0 

(4.) 
£      s.      a.     far. 

8542 
6 


38       6     4 


(3.) 

£    s.    d. 

10     7     4 

5 

51   16     8 


(5.) 

Ton.  c\vt.  qr.    Ibs.  oz. 
8       2       7     10 

7 


49     12 


0 


U 


6 


(6.) 


Ton.      cwt.      qr.        Ibs.        oz. 
4         15  3  8  5 

8 


38 


6 


16 


8 


cwt.        qr.      Ibs.         oz.        ar. 
9  1864 

9 

4  Ton.     4  cwt.     0  qr.     0  Ibs.     8  oz.     4  dr. 

(8.) 

cwt.          qr.       Ibs.         oz.         ar. 
10  3285 

10 

5  Ton.     7  cwt.     3  qr.     0  Ibs.     3  oz.     2  ar. 

30 


350 


INTEEMEDIATE   AKITHMETIC. 


(9.) 

(10.) 

m.   fur.   r.      yd.  ft. 

in.       m.   fur.    r.     y.     ft.    in. 

7       7     14       1       2 

1          234218 

11                                               12 

87       0     37       2       0 

5        29       5     13     3       0       6 

(11.) 

L.        m.     fur. 

r.         y.       ft.       in. 

325 

20         2         1         0 

14 

54         1          5 

5506 

(12.) 

(13.) 

m.   fur.   r.      y. 

ft,            Ib.  oz.  dwt.  gr. 

8       3     30       5 

2              1     10     14     20 

15                                   16 

127       0     25       2       1 

6            30       3     17       8 

(14.) 

(15.) 

Ibs.  oz.  dwt.  grs. 

Ibs.  oz.  dwt.  grs. 

2     11     12     15 

3     10     11     10 

17 

18 

39       5     14     15 

69     10       5     12 

(16.) 

(17.) 

Ibs.  oz.  dwt.  grs. 

sq.  m.  A.    R.     r.  yds. 

4       5     10       5 

1     40       2     30     20 

19 

20 

84       8     13     23 

21   173       3     13       6f 

MULTIPLICATION  OF   COMPOUND  NUMBERS.        351 

(18.)  (19.) 

sq.  m.  A.  R.  r.  sq.  m.  A.   R.  r.  yds. 

2  30   3  20       3  140   0   0  30 

21  22 


43   8   1  20  70  520   0  21  24-J 

(20.)  (21.) 

R.  r.  y.   ft.  in.  Tun.  pi.  hhd.  galls,  qt. 

1  20  20   8  T2  1   1   1  CO   2 

23  24 


34  35  27   8  108  47   1   1   3   0 

(22.)  (23.) 

pi.  hhd.  galls,  qt.   pt.  hhd.  galls,  qt.   pt.   gills. 

11     30       21  1     20       211 

25  26 


45       1       9       2       1  34     33       0       0       3 

(24.)  (25.) 

galls,     qrts.    pt.    gills,  day.     hr.     mi.     sec. 

40         2         1          2  1          3       20       40 

27  28 


1098    2    0 

2     31 

21   38   40 

(26.) 

(27.) 

(28.) 

hr.  ini.  sec. 

hr.  mi.  sec. 

hr.  mi.  sec. 

3  20  40 

2  20  30 

1  20  40 

29 

30 

31 

96     59     20  70     15       0  41     40     40 


352  INTERMEDIATE   AEITHMETIC. 

LESSON    XXXItV. 

3*50.    EXAMPLES. 

29.  If  a  family  consume  20  galls.  3  qts.of  wine  in 
one  week,  what  quantity  will   they  require  for  52 
weeks?  Ans.  17  hhd.  8  galls. 

30.  I  have  2  dozen  silver  spoons,  and  each  spoon 
weighs  13  oz.  7  dwt.  4  grs. ;  what  is  the  weight  of 
all  ?  Ans.  26  Ibs.  8  oz.  12  dwt.  0  grs. 

31.  Sam.  Jones  bought  7  squares  of  ground,  each 
square  measured  21   rods,  5   yds.  2  ft.  8  in. ;    how 
much  land  did  he  buy  ? 

Ans.  3  fur.  34  r.  2  yds.  2  ft.  2  in. 

32.  If  the  sun  moves  14  S.  0°  11'  30"  in  1  day, 
how  far  will  it  move  in  a  year  ? 

Ans.  426  C.  0  S.  9°  5V'  30" 

33.  If  1  dollar  will  buy  2  A.  1  R.  1  r.  1  yd.  1  ft. 
1  in.  of  ground,  how  much  land  can  I  buy  for  250 
dollars  ?         Ans.  564  A.  0  R.  19  r.  5  yds.  6  ft.  70  in. 

34.  If  a  man  can  travel  1  mi.  7  fur.  8  rd.  in  one 
hour,  how  far  can  he  travel  in  24  hours  ? 

Ans.  7  Lea.  0  mi.  4  fur.  32  rods. 

35.  A  load  of  hay  weighs  3  Tons,  1  cwt.  1  qr.  1 
lb.,  how  much  will  25  loads  weigh  ? 

Ans.  76  Ton.  11  cwt.  2  qr.  0  Ibs. 

36.  If  it  requires  12  yards,  2  qrs.  to  make  1  dress, 
how  many  yards  will  make  20  dresses  ?  Ans.  250  yds. 


MULTIPLICATION   OP   COMPOUND   NUMBERS.        353 

37.  If  1  boarder  consumes  2  Ibs.  1  oz.  3  drs.  of 
provisions,  how  much  will  26  boarders  consume  ? 

Ans.  2  qrs.  3  Ibs.  14  oz.  14  drs. 

38.  Multiply  2  yds.  1  qr.  1  na.  by  20. 

Ans.  60  yds.  2  nails. 

39.  If  1  dollar  will  buy  7  Ibs.  1  oz.  12  drs.  of  meat, 
how  much  can  I  buy  for  28  dollars  ?        Ans.  87  Ibs. 


LESSOR  XXXV. 

3«>1.    FRACTIONAL   PARTS. 

40.  Multiply  ^  mile  by  2. 

Ans.  5  fur.  13  r.  1  yd.  2  ft.  6  in. 

41.  Multiply  J-  mile  by  3.  Ans.  6  fur. 

42.  Multiply  J-  mile  by  3.  Ans.  3  fur. 

43.  Multiply  £  fur.  by  5.   ^4ws.  22  r.  1  yd.  0  ft.  8  in. 

44.  Multiply  TV  £  by  3.  Ans.  5s. 

45.  Multiply  £  £  by  5.  ^tws.  12s.  6d. 

46.  Multiply  ^  Tun  by  2.        ^ws.  1  pipe  42  galls. 

47.  Multiply  J  Tun  by  3.  ^tws.  1  pipe  1  hhd. 

48.  Multiply  .002  £  by  25.  Ans.  Is. 

49.  Multiply  .02  £  by  20.  Ans.  8s. 

50.  Multiply  .2s.  by  13.  Am.  2s.  7d.  T\  far. 

51.  Multiply  .025  mile  by  13.     Ans.  2  fur.  24  rds. 

52.  Multiply  .025  fur.  by  13.  Ans.  13  rods. 


354 


INTERMEDIATE   ARITHMETIC. 


53.  Multiply  .025  rod  by  15. 

Ana.  2  yds.  0  ft.  2.25  in. 

54.  Multiply  .03  gall,  by  12. 

Am.  1  qt.  0  pt.  3.52  gill. 

55.  Multiply  .003  hhd.  by  2. 

Am.  1  qt.  1  pt.  .096  gill. 

56.  Multiply  .08  League  by  20. 

Am.  1  L.  1  mi.  6  fur.  16  r. 

57.  Multiply  .008  League  by  25. 

Am.  4  fur.  32  rods. 

58.  Multiply  .018  Ton  by  12. 

Am.  4  cwt.  1  qr.  7  Ibs. 

59.  Multiply  .025  qr.  by  3.  Am.  1  Ib.  14  oz. 

60.  Multiply  .086  cwt.  by  4. 

Am.  1  qr.  9  Ibs.  6  oz.  6  drs.  TV 

61.  Multiply  .08  Ibs.  by  2.        Am.  2  oz.  8.96  drs. 

62.  Multiply  .025  £  by  3.  Am.  Is.  6d. 

63.  Multiply  .025  mile  by  30.         Am.  6  furlongs. 

64.  Multiply  .02  fur.  by  36. 

Am.  28  r.  4  yds.  1  ft.  24  in. 

65.  Multiply  .025  rod  by  38. 

Ans.  5  yds.  0  ft.  8.1  in. 

66.  Multiply  .0625  League  by  3. 

Ans.  4  fur.  20  rods. 

67.  Multiply  .08  League  by  200.  Ans.  16  Leagues. 


DIVISION   OF   COMPOUND   NUMBERS.  355 

DIVISION  OF  COMPOUND  LUMBERS. 

LESSON  XXXVI. 
35S.  Divide  2  £  3s.  4d.  and  3  far.  by  5. 

5)2  £  3s.  4d.  3  far.(8s.  8d.  0  far.  -§. 
2X20=40 

5)43 
40 

3X12  =  36 

5)40 
40 

5)3  far. 

5  into  2  £  cannot  be  divided,  so  that  we  must  re- 
duce the  pounds  to  shillings.  2  pounds  is  equal  to 
40  shillings,  and  3  shillings,  are  43  shillings  ;  5  into 
43  shillings,  8  times,  or  8  shillings,  and  3  shillings 
remaining.  Reduce  the  3  shillings  to  pence,  and 
add  the  pence  thus  found  to  the  pence  in  the  divi- 
dend— 3  shillings=36  pence  and  4  pence— 40  pence. 
5  into  40  pence,  8  times  or  8  pence,  and  0  remaining. 
5  into  3  far.,  0  farthings,  and  -|  remaining.  The 
answer  is  8s.  Sd.  Of  far. 

Or: 

Reduce  the  compound  number  to  the  lowest 
denomination.  Divide  the  number  found  by  the 


356 


INTEKHEDIATE  AEITHMETIC. 


given  divisor,  and  reduce  the  quotient  to  the  several 
higher  denominations. 

2  £  3s.  4d.  3  far.=2083  farthings. 
2083  far.  +  5=  41 6f  farthings. 

4)41 6| 

12)104  —  Of  far. 
8s.  8d. 


RULE. 

Divide  each  denomination  in  its  order,  as  in  sim- 
ple numbers,  beginning  with  the  highest.  If  there  is 
a  remainder,  reduce  the  remainder  to  the  next  lowest 
denomination,  and  add  in  the  number,  if  any,  of  this 
denomination  found  in  the  dividend.  Divide  as  be- 
fore. 

Continue  until  the  several  denominations  of  the 
dividend  are  divided.  The  quotient  thus  found  will 
be  the  answer. 


Or: 


353.  Reduce  the  dividend  to  its  lowest  denomi- 
nation. Divide  this  result  by  the  divisor,  and  reduce 
the  quotient  to  the  several  higher  denominations. 

(I   would   again  caution  the  pupil  in  regard 
Long  Measure  and  Square  Measure,  when  dividing 
by  fractional  parts.) 


DIVISION    OP   COMPOUND   NUMBERS.  357 

1.  Divide  3£  4s.  6d.  2  far.  by  3. 

Am.  l£ls.  6d.  §  far. 

2.  Divide  2£  5s.  8d.  3  far.  by  4. 

Ans.  11s.  5d.  I  far. 

3.  Divide  l£  6s.  4d.  2  far.  by  5. 

Ans.  5s.  3d,  1-J-  far. 

4.  Divide  3£  9s.  7d.  3  far.  by  6. 

Ans.  lls.  7d.  1£  far. 

5.  Divide  4£  8s.  lOd.  2  far.  by  7. 

.4ns.  12s.  8d.  If  far. 

6.  Divide  G£  2s.  Id.  3  far.  by  8. 

Ans.  15s.  3d.  ^  far. 

7.  Divide  4£  8s.  6d.  3  far.  by  9. 

Ans.  9s.  lOd.  I-  far. 

8.  Divide  2£  7s.  6d.  3  far.  by  10. 

Ans.  4s.  9d.  ^  far. 

9.  Divide  l£  2s.  3d.  2  far.  by  11. 

Ans.  2s.  Od.  l^j  far. 

10.  Divide  2£  3s.  4d.  2  far.  by  12. 

Ans.  3s.  7d.  l£  far. 

11.  Divide  17  hhds.  8  galls,  by  52. 

Ans.  20  galls.  3  quarts. 

12.  Divide  26  Ibs.  8  oz.  12  dwt.  0  grs.  by  24. 

Ans.  1  Ib.  1  oz.  7  dwt.  4  grs. 

13.  Divide  3  fur.  34  r.  2  y.  2  ft.  2  in.  by  7. 

Ans.  21  rods  5  yds.  2  ft.  8  in. 

14.  Divide  426  circ.  0  S.  9°  57'  30"  by  365. 

Ans.  14  S.  0°  11'  30". 

15.  Divide  564  A.  0  R.   19  r.  5  y.  6  ft.  70  in.  by 
250.  Ans.  2  A.  1  R.  1  rod  1  yd.  1  ft.  1  in. 


358  INTEEMEDIATE   AK1THMETIC. 

16.  Divide  7  lea.  0  m.  4  fur.  32  rods  by  24. 

Ans.  7  fur.  8  rods. 

17.  Divide  76  Ton  11  cwt.  2  qrs.  by  25. 

Ans.  3  Tons  1  cwt.  1  qr.  1  Ib. 

18.  Divide  250  yards  by  20.   Ans.  12  yards  2  qrs. 

19.  Divide  2  qrs.  3  Ibs.  14  oz.  14  drs.  by  26. 

Ans.  2  Ibs.  1  oz.  3  drams. 

20.  Divide  60  yards  2  nails  by  26. 

Ans.  2  yds.  1  qr.  1  na. 

21.  Divide  17£  14s.  4d.  by  8. 

Ans.  2£  4s.  3d.  2  far. 

22.  Divide  3  Tons  3  Ibs.  6  oz.  by  7. 

Ans.  8  cwt.  2  qrs.  7  Ibs.  10  oz. 

23.  Divide  38  Ton  6  cwt.  2  qr.  16  Ibs.  8  oz.  by  8. 

Ans.  4  Ton  15  cwt.  3  qrs.  8  Ibs.  5  oz. 

24.  Divide  4  Ton  4  cwt.  8  oz.  4  drs.  by  9. 

Ans.  9  cwt.  1  qr.  8  Ibs.  6  oz.  4  drs. 

25.  Divide  5  Ton  7  cwt.  3  qrs.  3  oz.  2  drs.  by  10. 

Ans.  10  cwt.  3  qr.  2  Ib.  8  oz.  5  drms. 

26.  Divide  87  mi.  37  rods  2  yds.  5  in.  by  11. 

Ans.  7  mi.  7  fur.  14  r.  1  yd.  2  ft.  1  in. 

27.  Divide  29  mi.  5  fur.  13  r.  3  y.  6  in.  by  12. 

Ans.  2  mi.  3  fur.  31  rods  0  y.  2  ft.  2  in. 

28.  Divide  54  Lea.   1    mi.  5   fur.  5  r.  5  yd.  6  in. 
by  14.  Ans.  3  Lea.  2  mi.  5  fur.  20  rods  2  yd.  1  ft.  0  in. 

29.  Divide  127  mi.  25  r.  2  yd.  1  ft.  6  in.  by  15. 

Ans.  8  mi.  3  fur.  31  r.  0  yd.  0  it.  6  in. 

30.  Divide  30  Ibs.  3  oz.  17  dwt.  8  grs.  by  16. 

Ans.  1  Ib.  10  oz.  14  dwt.  20  grs. 

31.  Divide  39  Ibs.  5  oz.  14  dwt.  15  grs.  by  17. 

Ans.  2  Ibs.  3  oz.  1 7  dwt.  7ff  grs. 


DIVISION  OF  COMPOUND  NUMBERS.  359 

32.  Divide  GO  Ibs.  10  oz.  5  dwt.  12  grs.  by  18. 

Ans.  3  Ibs.  10  oz.  11  dwt.  10  grs. 

33.  Divide  84  Ibs.  8  oz.  13  dwt.  23  grs.  by  19. 

Ans.  4  Ibs.  5  oz.  10  dwt.  5  grs. 

34.  Divide  34  R.  35  r.  27  y.  8  ft.  108  in.  by  23. 

Ans.  1  R.  20  rods  20  yds.  8  ft.  72  in. 

35.  Divide  1098  galls.  2  qts.  2  gills  by  27. 

Ans.  40  galls.  2  qts.  1  pt.  2  gills. 

36.  Divide  41  hrs.  40  mi.  40  sec.  by  31. 

Ans.  1  hr.  20  mi.  40  sec. 

37.  Divide  70  mi.  520  A.  21  r.  24J  yds.  by  22. 

Ans.  3  mi.  140  A.  0  R.  0  r.  30  yd. 

38.  Divide  34  hhd.  33  galls.  3  gills,  by  20. 

Ans.  1  hhd.  20  gall.  2  qts.  1  pt.  l-fa  gi- 

39.  Divide  70  hrs.  15  mi.  by  30. 

Ans.  2  hrs.  20  mi.  30  sec. 

40.  Divide  43  mi.  8  A.  1  R.  20  rd.  by  21. 

Ans.  2  mi.  30  A.  3  R.  20  rods. 

41.  Divide  45  pi.  1  hhd.  9  gall.  2  qt.  1  pt.  by  25. 

Ans.  1  pi.  1  hhd.  40  gall.  2  qt.  1  fc  pt. 

42.  Divide  96  hrs.  59  mi.  20  sec.  by  29. 

Ans.  3  hrs.  20  mi.  40  sec. 

43.  Divide  21  mi.  173  A.  3  R.  13  r.  6}  y.  by  20. 

Ans.  1  mi.  40  A.  2  R.  30  r.  20  yd. 
44.,Divide  47  Tun  1  pi.  1  hd.  3  gall,  by  24. 

Ans.  1  Tun  1  pi.  1  hhd  60  gall.  2  qt. 
45.  Divide  31  day  21  hrs.  38  mi.  40  sec.  by  28. 

Ans.  1  da.  3  hrs.  20  mi.  40  sec. 


360 


INTERMEDIATE   ARITHMETIC. 


LESSON  XXXVII. 
354.    EXAMPLES   IN   THE   FOREGOING   RULES. 

1.  What  cost  39  A.  2  R.  15  r.  of  land  at  $13.98 
per  acre.  Ans.  $553.52  + 

2.  What  cost  176  A.  3  R.  25  r.  of  land  at  $75.  per 
acre?  Ana.  $13267.96  + 

3.  What  cost  20  A.  2  R.  24  rods  of  land  at  $3  per 
acre  ?  Ans.  $61.95. 

4.  What  cost  10  yd.  3  qr.  2  na.  of  silk  at  80  cents 
per  yard  ?  Ans.  68.70. 

5.  What  cost  15  yd.  2  qr.  3  na.  of  cloth  at  $2.50 
per  acre?  Ans.  $39.21  +  . 

6.  What  cost  25  yd.  1  qr.  3  na.  silk  @  50  cents 
per  yard?  Ans.  $12.71+. 

7.  What  cost  67  bu.  3  pk.  7  qt.  potatoes  at  $2.50 
per  bushel  ?  Ans.  $169.92. 

8.  What  cost  125  bu.  3  pk.  1  qt.  of  wheat  at  87 
cents  per  bushel?  Ans.  $109.42. 

9.  What  cost  25  bushels  1  pk.  3  qt.  of  onions  at 
$2.50  per  bushel  ?  Ans.  $63.36. 

10.  What  cost  503  bu.  4  qts.  corn  at  43  cents  per 
bushel?  Ans.  $216.34. 

11.  What  cost  76  bu.  1  qt.  of  beans  at  66f  cents 
per  bushel  ?  Ans.  $50.68. 

12.  What  cost  10  bu.  3  pk.  of  pears  at  $1*50  per 
bushel?  Ans.  $16.125. 

13.  What  cost  25  bu.  1  pk.  of  corn  at  $1.35  per 
bushel  ?  Ans.  $34.08}. 


MISCELLANEOUS    EXAMPLES.  361 

14.  What  cost  1  bu.  1  pk.  1  qt.  peas  at  $1.50  per 
bushel?  Am.  $1.92  +  . 

15.  What  cost  17  cwt.  3  qrs.  23  Ibs.hay  at  $13.50 
per  Ton?  Am.  $1.11  +  . 

16.  What  cost  3  Ton  10  cwt.  3  qr.  of  iron  at  $30 
per  Ton?  Am.  $106.12$. 

17.  What  cost  16  boxes  sugar,  each  box  contain- 
ing 4  cwt.  3  qr.  18  Ibs.  at  $6  per  cwt.  ?  Am.  $473.28. 

18.  What  will  9  boxes  sugar  cost,  each  weighing 
8  cwt.  2  qr.  12  Ibs.,  at  812.50  per  cwt.  ?  Ans.  $969.75. 

19.  If  I  buy  16  cwt.  3  qr.  21  Ibs.  6  oz.  sugar,  at 
$7.50  per  cwt.,  what  will  my  bill  amount  to  ? 

Ans.  $127.22. 

20.  I  sold  5  cwt.  2   qr.  hay,  at  $27.50  per  Ton. 
What  was  the  amount  of  my  sale?  Ans.  $7.36$. 

21.  If  I  can  ride  5  fur.  3  r.  10  ft.  6  in.  in  a  steam- 
car  for  22  cents  per  mile,  what  ought  to  be  my  fare  ? 

A?ts.  14  cents. 

22.  To  every  one  of  my  10  children  I  gave  26£ 
14s.  8d.  3  far.     How  much  did  all  of  them  receive? 

Ans.  2£  13s.  5d.  2Jff  far. 

23.  If  I  can  travel  1  mile  1  fur.  1  rod  1  ft.  1  in.  in 
one  day,  how  far  can  I  travel  in  20  days  ? 

Am.  22  mi.  4  fur.  21  rods  1  yd.  1  ft.  2  in. 

24.  Add  together  4  feet  3  in.          Ans.  51  inches. 

25.  Reduce  each  of  the  following  to  a  compound 
number:  3.250£,  32.5£,  325. 2£.      Last  Ans.  325£  4s. 

26.  Reduce  1  hour  to  the  decimal  of  a  day. 

Ans.  0.0416f. 

27.  Reduce  1  farthing  to  the  decimal  of  a  <£. 

Ans.  .001041+. 


362 


INTERMEDIATE  ARITHMETIC. 


28.  Reduce  1  foot  to  the  decimal  of  a  yard. 

Ans.  0.33J. 

29.  Reduce  1  cent  to  the  decimal  of  a  dollar. 

Ans.  .01. 

30.  Reduce  1  shilling  to  the  decimal  of  a  £. 

Ans.  .05£. 

31.  How  many  inches  in  1  mile?         Ans.  63360. 

32.  How  many  inches  in  1  square  mile  ? 

Ans.  4014489600. 

33.  How  many  miles  in  2534400  inches  ? 

Ans.  40  miles. 


CHAPTER  XV. 

DUODECIMALS. 
LESSON"  I. 

35«>.  DUODECIMALS  increase  and  decrease  in  a 
twelvefold  ratio.  The  table  is  uniform,  it  requiring 
12  of  a  lower  denomination  to  make  1  of  a  higher. 
The  highest  denomination  is  the  foot,  then  comes 
primes,  seconds,  thirds,  fourths,  etc.,  etc. 

356.  The  number  of  commas  above  the  figure 
shows  its  denomination.  Thus,  4'  is  read  4  primes, 
3"  is  read  3  seconds,  8'"  is  read  8  thirds,  9""  is  read 
9  fourths,  etc. 


DUODECIMALS.  363 


TABLE. 

12  thirds  ('")  make  1  second. 

12  seconds          "      1  prime. 

12  primes  "      1  foot.  ft. 

357.  In  multiplication,  the  indices  are  added, 
and  written   over  the  product.     Thus,  G"X3'=18'". 
That  is,  6  seconds  multiplied  by  3  primes  is  equal  to 
18  thirds. 

358.  In  division,  the  index  of  the  divisor  is 
subtracted  from  the  index  of  the  dividend.     Thus, 
18"'-^6"=3'.     That  is,  18  thirds  divided  by  6  sec- 
onds is  equal  to  3  primes. 

359.  NOTE. — The  above  table  may  be  carried  out  as  far  as 
the  pupil  may  in  his  judgment  see  proper.  The  name  given  to  the 
other  denominations  is  determined  by  the  number  of  strokes  or 
commas  written  over  it.  Thus,  6'"""  is  read  6  sevenths,  8'""  ""is 
read  8  tenths,  etc.  In  calculating,  it  will  be  correct  enough  to 
bring  the  answer  to  thirds.  One  third  is  nVs  Paft 
which  is  small  enough  for  any  answer. 


LESSON  II. 

3 GO.      ADDITION    OF   DUODECIMALS. 

For  addition  of  Duodecimals  the  same  rule  ap- 
plies as  for  addition  of  denominate  numbers.  The 
scale  is  uniform,  every  higher  denomination  requir- 
ing 12  of  a  lower  one  up  to  feet.  The  table  then 
corresponds  to  Long  measure. 


364 


INTERMEDIATE    ARITHMETIC. 


EXAMPLE. 

Add  2  ft. 

3' 

4" 

6  ft. 

8 

9 

12  ft. 

3 

8 

21 

4 

1 

Adding  the  first  column  of  seconds  we  find  25. 
12  seconds  make  1  prime.  12  into  25,  2  times  and 
1  sec.  remaining.  Write  the  1  sec.  under  the  column 
of  seconds,  and  carry  the  2  primes  to  the  column  of 
primes.  Adding  the  second  column,  gives  14,  and 
2 'we  had  to  carry,  makes  16.  12  primes  make  one 
foot.  12  into  16,  once  and  4  over.  Write  the  4  re- 
maining primes  under  the  column  of  primes,  and 
carry  1  to  the  next  column.  Adding  the  column  of 
feet  gives  20  feet,  and  1  foot  makes  21  feet.  Write 
the  sum  under  the  column  of  feet,  and  the  required 
answer  will  be  21  ft.  4'  1". 


361.  Masons  and  artists  use  this  measure,  and 
calculate  their  work  by  the  foot,  prime  and  second. 
The  yard  is  very  seldom  make  use  of,  except  in  cases 
where  work  is  taken  by  the  yard,  such  as  paving 
streets,  building  stone  bridges,  etc.,  etc. 


ADDITION   OF   DUODECIMALS.  365 


EXAMPLES. 


(1) 

(2) 

(3) 

ft. 

' 

" 

ft. 

' 

" 

ft. 

' 

" 

2 

4 

6    * 

1 

1 

1 

4 

4 

4 

4 

5 

9 

2 

2 

2 

2 

6 

8 

8 

7 

6 

3 

3 

3 

3 

7 

9 

4 

3 

2 

4 

4 

4 

4 

8 

7 

9 

8 

7 

5 

5 

5 

5 

9 

6 

8 

6 

4 

6 

6 

6 

21 

0 

10 

37 

11 

10 

"7 

7 

7 

30         6         4 
(4)  (5)  (6) 


ft. 

' 

" 

ft. 

' 

" 

ft. 

' 

" 

3 

2 

8 

4 

9 

3 

6 

7 

8 

1 

4 

3 

3 

9 

4 

9 

4 

3 

9 

7 

6 

9 

4 

3 

2 

8 

7 

8 

2 

4 

3 

4 

9 

4 

3 

2 

6 

4 

8 

2 

1 

2 

9 

8 

4 

28         9         5  23         4         9  32 


(7)  (8)  (9) 

ft.          '         "  ft.  "  ft. 


5 

8 

3 

7 

2 

4 

1 

4 

6 

2 

7 

9 

2 

9 

9 

2 

3 

4 

6 

4 

3 

8 

7 

6 

5 

6 

7 

9 

8 

7 

3 

8 

4 

8 

9 

4 

4 

3 

2 

9 

8 

7 

3 

8 

2 

28         8         0  32         0         6  21          7       11 


366 


INTERMEDIATE  ARITHMETIC. 


(10) 


(11) 


(12) 


ft. 

/ 

n 

5 

3 

8 

4 

6 

9 

2 

8 

7 

9 

5 

4 

13 

12 

G 

36 

0 

10 

(13) 

ft. 

/ 

// 

7 

4 

3 

2 

9 

8 

5 

3 

8 

7 

2 

9 

8 

4 

3 

(16) 

ft. 

/ 

n 

2 

8 

7 

3 

4 

6 

9 

8 

7 

2 

4 

9 

8 

7 

3 

ft. 

' 

u 

4 

6 

2 

2 

4 

6 

8 

9 

7 

3 

4 

1 

9 

8 

2 

(14) 

v 

ft. 

'    • 

rr 

1 

3 

6 

2 

8 

9 

7 

4 

3 

9 

8 

7 

G 

8 

9 

ft. 

(17) 

4 

2 

3 

3 

6 

9 

1 

2 

3 

2 

4 

8 

7 

6 

4 

ft. 

/ 

// 

17 

3 

8 

9 

2 

4 

4 

3 

1 

8 

7 

6 

2 

4 

8 

9 

3 

1 

(15) 

ft. 

/ 

n 

2 

6 

8 

9 

4 

9 

2 

1 

8 

9 

7 

8 

8 

1 

9 

(18) 

ft. 

/ 

a 

7 

2 

6 

2 

9 

8 

3 

4 

7 

2 

8 

4 

7 

6 

3 

(19) 

ADDITION   OF  DUODECIMALS. 
(20) 

(21) 

367 

ft. 

t 

" 

ft. 

A 

" 

ft. 

' 

" 

7 

4 

2 

17 

2 

9 

8 

7 

6 

3 

2 

1 

3 

8 

7 

1 

2 

3 

2 

8 

7 

2 

9 

4 

4 

5 

6 

9 

4 

3 

1 

2 

3 

7 

8 

9 

8 

10 

11 

4 

5 

G 

10 

11 

10 

(22)  (23)  (24) 

ft.         '         "  ft.         '         "  -  ft. 


4 

5 

6 

2 

4 

6 

2 

8 

9 

7 

8 

9 

2 

8 

6 

9 

4 

3 

10 

11 

10 

4 

9 

7 

3 

2 

8 

9 

8 

7 

4 

3 

9 

9 

7 

6 

(25)  (26)  (27) 

ft.         '         "  ft.         '         "  ft. 


4 

8 

7 

6 

8 

7 

7 

9 

10 

6 

4 

3 

2 

9 

8 

1 

2 

3 

9 

8 

2 

1 

9 

9 

4 

5 

6 

4 

6 

1 

3 

4 

6 

7 

8 

9 

10       11       11 


368 


INTERMEDIATE   ARITHMETIC. 


(28) 


(29) 


(30) 


ft. 

/ 

n 

2 

4 

8 

1 

2 

4 

4 

2 

1 

6 

7 

8 

7 

8 

6 

(31) 

ft. 

/ 

a 

7 

6 

3 

2 

8 

7 

2 

4 

9 

12 

8 

7 

6 

4 

3 

(34) 

ft. 

/ 

// 

2 

6 

8 

9 

8 

12 

12 

1 

3 

4 

2 

6 

6 

8 

9 

ft. 

/ 

n 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

1 

2 

3 

(32) 

ft. 

/ 

n 

8 

9 

3 

2 

4 

6 

3 

6 

9 

1 

2 

3 

1 

3 

2 

(35) 

ft. 

/ 

n 

8 

7 

6 

2 

4 

6 

8 

9 

4 

2 

6 

3 

1 

4 

5 

ft. 

/ 

// 

9 

8 

7 

C 

5 

4 

1 

2 

3 

3 

2 

1 

9 

8 

9 

(33) 

ft. 

/ 

n 

9 

8 

7 

2 

1 

2 

3 

1 

4 

4 

1 

5 

5 

1 

6 

(36) 

ft. 

/ 

a 

9 

8 

7 

2 

9 

3 

8 

7 

9 

12 

2 

8 

4 

1 

9 

ADDITION    OF   DUODECIMALS.  369 


(37) 

(38) 

(39) 

ft. 

' 

" 

ft. 

' 

n 

ft. 

' 

" 

7 

2 

3 

3 

8 

7 

1 

1 

1 

1 

2 

9 

2 

1 

4 

2 

2 

2 

1 

8 

G 

1 

9 

7 

3 

3 

3 

1 

4 

3 

6 

2 

1 

5 

5 

5 

1 

6 

3 

8 

9 

7 

4 

4 

4 

6 

6 

6 

7 

7 

7 

8 

8 

8 

(40) 

(41) 

(42) 

ft. 

' 

n 

ft. 

i 

" 

ft. 

' 

" 

2 

3 

6 

7 

6 

9 

1 

2 

3 

2 

3 

6 

9 

6 

7 

4 

5 

6 

2 

3 

6 

6 

7 

9 

7 

8 

9 

2 

3 

6 

9 

7 

6 

10 

11 

1 

2 

3 

6 

2 

4. 

8 

2 

3 

4 

2 

3 

6 

8 

2 

4 

5 

G 

7 

2 

3 

6 

1 

2 

3 

8 

9 

10 

2 

3 

6 

11 

1 

2 

(43) 

ft. 

' 

it 

1 

2 

6 

1 

2 

3 

6 

5 

4 

7 

8 

9 

3 

2 

1 

f 

4 

5 

6 

9 

8 

7 

370 


INTEKMEDIATE  ARITHMETIC. 


LESSON     III. 
363.   SUBTRACTION   OF   DUODECIMALS. 

In  subtraction  of  duodecimals  the  same  rule  ap- 
plies as  in  subtraction  of  denominate  numbers. 

EXAMPLE. 

From  24  ft.  8'  2"  take  8  ft.  9'  4". 


OPEEATION. 

24  ft.     8'     2" 
8          94 


15 


10    10 


Begin  at  the  right  hand  col- 
umn and  subtract.  4"  from  2", 
you  cannot  subtract,  borrow  1', 
which  is  equal  to  12".  Add  the 
12"  to  the  2",  making  14".  4" 
from  14"  leaves  10".  Write  the  10"  under  the  col- 
umn, and  return  the  1'  you  borrowed.  Add  it 
to  the  subtrahend,  which  now  becomes  (9'  +  !')  10'. 
Subtract  the  second  column.  10'  from  8'  you  cannot 
subtract.  Borrow  1  foot,  which  is  equal  to  12'. 
Add  the  12'  and  the  S\  12' +  8=20.  Now  sub- 
tract. 10'  from  20'  leaves  10'.  Write  the  10'  under 
the  column  of  primes,  and  add  1  ft.  to  the  subtrahend 
in  the  column  of  feet.  1  ft.  and  8  ft.  are  9  ft.  9  ft. 
from  24  ft.  leaves  15  ft.  The  required  answer  is  15 
ft.  10'  10". 


1.  From 
Take 


EXAMPLES. 

14  ft.  8' 

10  ft.  3' 


6" 
4" 


SUBTRACTION    OF   DUODECIMALS.  371 


2. 

From 

28ft. 

9' 

8" 

Take 

14  ft. 

8' 

7" 

13 

9 

8 

3. 

From 

1C  ft. 

9' 

8" 

Take 

3  ft. 

8' 

9" 

13 

0 

11 

4. 

From 

86  ft. 

4' 

9" 

Take 

78ft. 

G' 

10" 

7 

9 

11 

5. 

From 

16ft. 

3' 

8" 

Take 

2  ft. 

4' 

9" 

13 

10 

11 

6. 

From 

20  ft. 

4' 

6" 

Take 

18  ft. 

5' 

8" 

1 

10 

10 

7. 

From 

22  ft. 

G' 

8" 

Take 

21  ft. 

7' 

9" 

10 

11 

8. 

From 

34ft. 

6' 

2" 

Take 

33  ft. 

7' 

3" 

10 

11 

9. 

From 

18ft. 

2' 

1" 

Take 

4ft. 

3' 

2" 

13  10  11 


372 


INTERMEDIATE   ARITHMETIC. 


10. 

From 

18ft. 

9' 

7" 

Take 

12  ft. 

10' 

8" 

5 

10 

11 

11. 

From 

26  ft. 

4' 

6" 

Take 

18  ft. 

3' 

.9" 

12. 

From 

16ft. 

10' 

8" 

Take 

14  ft. 

9' 

8" 

13. 

From 

22  ft. 

6' 

8" 

Take 

18ft. 

5' 

9" 

14. 

From 

18ft. 

4' 

3" 

Take 

12  ft. 

9' 

4" 

15. 

From 

27ft. 

5' 

8" 

Take 

26  ft. 

6' 

9" 

16. 

From 

16  ft. 

4' 

8" 

Take 

10ft. 

3' 

9" 

17. 

From 

23  ft. 

4' 

8" 

Take 

22  ft. 

5' 

9" 

18. 

From 

28ft. 

r 

9" 

Take 

18ft. 

9' 

8" 

19. 

From 

16  ft. 

3' 

8" 

Take 

15  ft. 

4' 

7" 

SUBTRACTION   OF  DUODECIMALS.  373 


20.  From 
Take 

21.  From 
Take 

22.  From 
Take 

23.  From 
Take 

24.  From 
Take 

25.  From 
Take 

26.  From 
Take 

27.  From 
Take 

28.  From 
Take 

29.  From 
Take 

30.  From 
Take 

291  ft. 
17ft. 

8' 
9' 

6" 

8" 

19ft. 
18  ft. 

8' 
9' 

3" 

2" 

18ft. 
14  ft. 

8' 
8' 

9"          10'" 
10"            9'" 

16ft. 
14  ft. 

2' 
8' 

4" 
9" 

2  ft. 
1ft. 

3'     8" 
4'     6" 

Htll          rtlttt         Htllll 

9'"     7""     8"'" 

16  ft. 
12  ft. 

4' 
6' 

3" 
8" 

6  ft. 
2  ft. 

4' 
3' 

7"         o'"          0"" 
2"      -    8'"           9"" 

24ft. 
13  ft. 

8' 
4' 

7" 
8" 

17ft. 
3ft. 

2' 
4' 

6"         7'"          8"" 
8"          9'"          8"" 

16  ft. 
12  ft. 

3' 
4' 

8" 
9" 

16  ft. 
4ft. 

3' 
5' 

8"         7'"          9"" 
6"         9'"        10"" 

374 


INTERMEDIATE  AKITHMETIC. 


LESSON  IV. 
MULTIPLICATION   OF   DUODECIMALS. 

The  same  rule  applies  as  for  multiplication  of 
denominate  numbers. 

EXAMPLE. 
Multiply  3  ft.  4'  6"  by  8. 

Multiply  the  seconds.  8  X  6"= 
48".  12"=l'.  12  into  48, 4  times 
and  0"  remaining.  Write  the  0" 
under  the  seconds  and  carry  the  4 
primes.  Multiply  the  primes. 
8X4' =  32'  and  4'  are  36'.  12' 
make  1  foot.  12  into  36,  3  times  and  0'  remaining. 
Write  the  0'  under  the  primes  and  carry  the  3  feet. 
Multiply  the  feet.  8X3=24,  and  3  are  27  ft.  Write 
27  under  the  column  of  feet.  The  answer  is  27  feet  0 
primes  0  seconds. 


OPEEATION. 

3ft.    4'    6" 
8 

27  ft.    0'    0" 


EXEECISES. 

1.  Multiply  2  ft.  4'  6"  by  3. 

2.  Multiply  3  ft.  5'  3"  by  4. 

3.  Multiply  4  ft.  6'  4"  by  5. 

4.  Multiply  5  ft.  7'  5"  by  6. 

5.  Multiply  6  ft.  8'  6"  by  7. 

6.  Multiply  7  ft.  9'  7"  by  8. 

7.  Multiply  8  ft.  10'  8"  by  9. 

8.  Multiply  9  ft.  11'  9"  by  3. 

9.  Multiply  10  ft.  1'  10"  by  4. 


Am.  7  ft.  1'  6". 

Am.  13  ft.  9'  0". 

Am.  22  ft.  7'  8". 

Ans.  33  ft.  8'  6". 
Am.  46  ft.  11'  6". 

Am.  62  ft.  4'  8". 

Am.  80  ft.  0'  0". 
Am.  29  ft.  11'  3". 

Am.  40  ft.  7'  4". 


MULTIPLICATION   OF  DUODECIMALS.  375 

10.  Multiply  11  ft.  2'  11"  by  5.     Am.  56  ft.  2'  7". 

11.  Multiply  12  ft.  3'  2"  by  6.       Ans.  73  ft.  7'  0". 

12.  Multiply  13  ft.  4'  3"  by  7.       Ans,  93  ft.  5'  9". 

13.  Multiply  14  ft.  5'  4"  by  8.     ^wa.  115  ft.  6'  8". 

14.  Multiply  15  ft.  6'  5"  by  9.     Ans.  139  ft.  9'  9". 

15.  Multiply  16  ft.  7'  6"  by  10.  Ans.  166  ft.  3'  0". 

16.  Multiply  17  ft.  8'  7"  by  11. 

Ans.    194  ft.  10'  5". 

17.  Multiply  18  ft.  9'  8"  by  12.  Ans.  225  ft.  8'  0". 


LESSON  V. 

CASE   II, 
864.   TO   FIND   THE    AKEA  OF   ANT   PARALLELOGRAM. 

RULE. 
Multiply  the  length  by  the  breadth. 

EXAMPLE. 

1.  Find  the  area  of  a  room  18  ft.  4'  3"  long  and 
12  ft.  6'  4"  wide. 

OPERATION. 


18ft. 

4' 

3" 

12  ft. 

6' 

4" 

72" 

16'" 

12'" 

108' 

24" 

18'" 

48' 

36" 

216 

229         11'         2"         10' 


376 


INTERMEDIATE   ARITHMETIC. 


Analyse  as  follows:  Seconds  by  seconds  gives 
("")  fourths.  4  X  3=12"".  Seconds  by  primes  gives 
('")  thirds.  4  X  4=1 6.  Seconds  by  feet  give  seconds. 
4"X  18  ft.=72.  Multiply  by  the  6'.  Primes  by  sec- 
onds gives  ("')  thirds.  6'X3"=18'".  Write  the 
18"'  under  the  16"'.  6'  by  4'=24"  seconds.  Write 
the  24"  under  the  72".  6'XIS  ft.=108'.  Multiply 
by  the  12  ft.  Feet  by  seconds  gives  seconds.  12 
ft  x  3"=36".  Write  the  36  under  the  column  of  sec- 
onds. Feet  by  primes  gives  primes.  12X4'=48'. 
Write  the  48'  under  the  column  of  primes.  Feet  by 
feet  give  square  feet  12  X  18=216  feet.  Add  the  sev- 
eral products  beginning  at  the  12"".  12""  make  I'". 
12  into  12  once  and  0""  remaining.  Add  the  thirds. 
16  + 18=34'".  12=1.  12  into  34  2  times  and  10'", 
over.  Write  the  10'"  under  the  column  of  thirds. 
Add  the  column  of  seconds.  72  +  24  +  36=132  and  2 
to  carry  makes  134".  12=1.  12  into  134  11  times 
and  2"  over.  Write  the  2"  under  the  column  of  sec- 
onds. Add  the  column  of  primes.  108  +  48=156 
and  11=167.  12'  make  1  foot.  12  into  167  13  times 
and  11'  remaining.  Write  the  11'  under  the  column 
of  primes  and  carry  the  13  to  the  next  column.  216 
+  13=229  square  feet.  The  answer  is  229  sq.  ft.  11' 
2"  10'". 


EXERCISES. 


1.  A  school-room  is  28  ft.  4'  in  length,  and  18  ft. 
3'  in  breadth.  How  many  square  feet  of  flooring  is 
required  for  the  room  ?  Am.  517  ft.  1'. 


MULTIPLICATION   OF   DUODECIMALS.  377 

2.  If  the  above  flooring  is  worth  124-  cents  a  foot, 
how  much  would  it  cost  ?  Ans.  $64.33  -f . 

3.  How  much  would  it  cost  to  plaster  the  ceiling 
of  that  room  at  25  cents  per  foot  ?        Ans.  $129.20  + . 

4.  A  ball-room  is  108  ft.  3'  in  length,  and  84  ft. 
3'  in  breadth.     How  many  square  feet  of  plastering 
would  be  required  to  cover  the  ceiling  ? 

Ans.  9120  ft.  0'  9". 

5.  How  much  would  the  above  plastering  cost, 
at  30  cents  a  foot  ?  Ans.  $2736.002  + . 

6.  If  the  room  was  40  feet  high,  how  many  ft.  of 
plastering  would  be  required  to  cover  the  4  sides, 
providing  there  were  no  openings  ?       Ans.  15400  ft. 

7.  How  much  would  the  plastering  of  the  ceiling 
and  the  4  sides  cost,  at  $1  per  square  yard? 

Ans.  $2724.44  +  . 

8.  Suppose  there  were  6  openings  on  each  side  of 
the  ball-room,  4  feet  wide  and  9  ft.  3' high;  how 
many  yards  of  plastering  would  be  required  to  cover 
the  ball-room?  Ans.  2625  yds.  7  ft.  6'  9". 

9.  How  much  would  the  plastering  of  the  room 
cost,  less  the  openings,  @  $1.25  per  square  yard  ? 

Ans.  $3282.221+. 

10.  A  yard  is  200  feet  long,  by  48  feet  wide ; 
how  many  square  feet  in  that  yard  ?       Ans.  9600  ft. 

11.  How  many  bricks  would  cover  that  yard,  al- 
lowing 40  bricks  to  the  square  yard  ?     How  much 
would  the  bricks  cost  at  $12^  per  thousand? 

1st  Ans.  42566f  bricks.     2d  Ans.  $532.08. 


378  INTERMEDIATE   AEITHMETIC. 

12.  My  fence  is  200  feet  long  by  12  feet  high. 
How  many  square  feet  of  lumber  in  my  fence,  and 
what  is  it  worth  at  $13  per  thousand  feet  ? 

Ans.  $31.20 

13.  A  plantation  has  a  front  of  28  acres  by  a 
depth  of  200  acres.     How  many  acres  in  that  planta- 
tion? Ans.  5600  acres. 

14.  A  mason  undertakes  to  build  the  walls  of  a 
fort  for  $33J  per  square  yard.     The  fort  is  690  ft.  8' 
square,  and  10  ft.  high.     How  much  is  his  contract  ? 

Ans.  $102320.987+. 

15.  A  brick  is  8  inches  long  and  4  inches  wide. 
How  many  bricks  do  I  require  to  pave  a  sidewalk 
300  ft.  long  by  12  ft.  wide  ?  Ans.  16200  bricks. 

16.  How  many  bricks  do  I  require  to  pave  a  yard 
90  feet  long  and  40  feet  wide  ?  Ans, 

17.  A  city  railroad  is  2-J-  miles  long.     It  is  re- 
quired to  pave  12  feet  in  width  across  the  track. 
How  much  would  the  paving  cost  at  45  cents  a 
yard  ?    If  the  stringers  of  each  track  were  8  inches 
wide,  how  many  yards  should  be  deducted  from  the 
superficial  measurement  ? 

$7040,  1st  Ans.     1955f  2d  Ans. 

LESSON   VI. 

«IG»"5.   DIVISION   OF   DUODECIMALS. 

1.  Divide  34  ft.  5'  6"  by  3.  Ans. 

Reduce  feet  and  primes  to  seconds  : 
12  x  34=408,  408+5=413' 
12  x  413'=4956,  4956+6=4962". 


DIVISION   OF  DUODECIMALS.  379 

OPERATION. 

4962-;-3=lG54",  reduced  gives  11  ft.  5'  10" 
Or: 

Divide  as  follows : 

3)34  ft.  5'  6"(H  ft.  5'  10' 


1 
12 

12 
5 

17 
15 


2 
12 

24 
6 

30 
30 


Analysis  of  the  above  sum : 

3  into  34  ft.,  11  times,  11  times  3  are  33,  33  from 
34  leaves  1  foot.  1  foot  reduced  to  primes  is  equal 
to  12X1=12',  and  5',  are  17  primes.  3  into  17 
primes,  5  times,  5  times  3  are  15,  15  from  17  leaves 


380 


INTERMEDIATE   ARITHMETIC. 


2.     2  primes   reduced   to  seconds   gives   2x12=24 
seconds,  and  6  seconds  make  30  seconds.     3  into  30, 
10  times,  10  times  3  are  30,  30  from  30  leaves  0. 
The  answer  is  11  ft.  5'  10". 


EXERCISES. 

1.  Divide  14  ft.  6'  8"  by  2. 

Ana.  7  ft.  3'  4". 

2.  Divide  21  ft.  9'  6"  by  3. 

Ans.  7  ft.  3'  2". 

3.  Divide  16  ft.  4'  8"  by  4. 

Ana.  4  ft.  1'  2". 

4.  Divide  20  ft.  6'  8"  by  5. 

Ans.  4  ft.  1'  4". 

5.  Divide  36  ft.  6'  6"  by  6. 

Ans.  6  ft.  1'  1". 

6.  Divide  42  ft.  7'  8"  by  7. 

Ana.  6  ft.  1'  If. 

7.  Divide  16  ft.  8'  8"  by  8, 

Ans.  2  ft.  1'  1". 

8.  Divide  27  ft.  9'  9"  by  9. 

Ans.  3  ft.  1'  1". 

9.  Divide  20  ft.  10'  10"  by  10. 

Ana.  2  ft.  1'  1". 

10.  Divide  25  ft.  11'  11"  by  11. 

Ans. 

11.  Divide  24ft.  8'  8"  by  12. 

Ans. 

12.  Divide  26  ft.  3'  3"  by  13. 

Ans. 

13.  Divide  14  ft.  6'  6"  by  14. 

Ans. 

14.  Divide  15  ft.  4'  4"  by  15. 

Ans. 

15.  Divide  48  ft.  3'  3"  by  16. 

Ans. 

16.  Divide  34ft,  6'  6"  by  17. 

Ans. 

17.  Divide  36  ft.  4'  4"  by  18. 

Ans. 

18.  Divide  38  ft.  5'  6"  by  19. 

Ans. 

19.  Divide  44  ft.  6'  6"  by  20. 

Ans. 

20.  Divide  38  ft.  6'  6"  by  21. 

Ans. 

21.  Divide  44  ft.  8'  8"  by  22. 

Ans. 

22.  Divide  46  ft.  6'  6"  by  23. 

Ans. 

23.  Divide  69  ft.  3'  3"  by  24. 

Ans. 

24.  Divide  75  ft.  4'  4"  by  25. 

Ans. 

25.  Divide  98  ft.  3'  3"  by  26. 

Ans. 

DIVISION   OF   DUODECIMALS. 

2G.  Divide  54  ft.  6'  0"  by  27.  Ans. 

27.  Divide  56  ft.  8'  8"  by  28.  Ans. 

28.  Divide  68  ft.  4'  4"  by  29.  Ans. 

29.  Divide  90  ft.  6'  8"  by  30.  Ans. 

30.  Divide  93  ft.  6'  9"  by  31.  Ans. 

31.  Divide  24  ft.  8'  8"  by  32.  Ans. 

32.  Divide  20  ft.  6'  6"  by  33.  Ans. 

33.  Divide  2  ft.  1'  2"  by  34.  Ans. 


LESSON  VII. 

366.  The  area  and  one  of  the  sides  of  a  square 
•figure  being  given,  to  find  the  other  side. 

RULE. 
Divide  the  area  by  the  given  side. 

EXAMPLE. 

1.  The  area  of  my  farm  is  64000  square  feet,  and 
the  length  of  one  of  its  sides  is  450  ft.  2  primes  6 
seconds.  What  is  the  length  of  the  other  side  ? 

Arrange  the  numbers  as  for  division  of  denomi- 
nate numbers,  and  divide  the  highest  term  of  the 
dividend  by  the  highest  term  of  the  divisor.  Then 
multiply  all  of  the  terms  of  the  divisor  by  the  quo- 
tient thus  obtained,  subtract  the  result  from  the 
corresponding  terms  of  the  dividend,  and  take  down 
the  next  denomination  of  the  dividend  to  form  a  new 
divisor.  It  may  happen  that  the  highest  term  of  the 
divisor  may  be  too  great  for  the  highest  term  of  the 
dividend.  If  so,  reduce  the  highest  term  of  the  div- 


382 


INTERMEDIATE   ARITHMETIC. 


idend  to  the  next  lower  denomination.  Divide  as 
before,  care  being  taken  to  place  the  proper  sign  to 
the  quotient  figure. 

NOTE. — In  division,  the  indices  of  the  divisor  are  subtracted 
from  those  of  the  dividend,  and  the  remainder  will  show  what  index 
belongs  to  the  quotient  figure. 

3l>7.  Thus,  primes  divided  by  primes  give  primes,  seconds 
divided  by  seconds  give  seconds,  seconds  divided  by  primes  give 
primes,  thirds  divided  by  primes  give  seconds,  etc.,  etc.,  etc.,  etc., 
etc.,  etc. 

OPERATION. 

450  ft.  2'  6")6400  ft.  0'  0"(14  ft.  0'  4"  4'"  9"" 
6302     11    0 


97  ft.  1'  0" 
165'  0"  0"' 

1980"  0'"  0"" 
1800"  10'"  0"" 

180"    2'"  0"" 
2162'"  0""  0'"" 
1800'"  10""  0""' 


361' 


0'" 


4334""  0""'  0"" 
4051""  10'""  6""" 

272""     1""'  Q""u 
etc.,  etc. 

ANALYSIS. 

450  ft.  2'  6"  into  6400  ft.  0'  0"  14  times  (14  feet). 
14  x  450  ft.  2'  6"  =  6302  ft.  11'  0".  Subtracting  6302 
ft.  11'  0"  from  the  dividend  leaves  a  remainder  of  97 


DIVISION   OF   DUODECIMALS.  383 

ft.  1'  0".  Reducing  97  ft.  1'  10"  to  primes  gives  165 
primes  0'  0'".  450  ft.  2'  6"  into  1G5  primes,  \vo  can- 
not divide.  Reduce  105  primes  to  seconds  and  write  0 
primes  in  the  quotient.  165  primes  is  equal  to  1980" 
450  ft.  2'  6"  into  1980",  4"  times.  4  x  450  ft.  2'  6"  = 
1800"  10'"  0"".  Subtracting  1800"  10'"  0""  from  1980" 
leaves  180"  2"'  0"".  Reducing  180"  2'"  to  thirds 
gives  2162"'  450  ft.  2'  6"  into  2162'",  4'"  times.  4  x 
450  ft.  2'  G"=1800'"  10""  0'"".  Subtracting  leaves 
361"'  2""  0'"".  Reducing  361'"  2""  to  fourths  gives 
4334""  0'"".  Dividing  4334""  by  450  ft,  2'  6",  gives- 
9""  for  the  quotient.  9""x450  ft.  2'  G"  =  4051"" 
10""'  6""".  Subtracting,  leaves  272""  1'""  6""". 

This  operation  may  be  carried  out  as  far  as  the 
pupil  sees  fit. 

EXAMPLES. 

1.  A  piece  of  carpet  contains  517  ft.   1'.    The 
length  of  one  of  its  sides  is  18  ft.  3'.     What  is  the 
length  of  the  other  side?  Ans.  28  ft.  4' 

2.  The  flooring  of  a  room  contains  9120  ft.  0'  9", 
and  it  is  84  ft.  3'  in  width.    How  long  is  the  room? 

Ans.  108  ft.  3'. 

3.  My  gallery  is  28  ft.  4'  long  and  has  an  area  of 
517  ft.  1'.     How  wide  is  it?  Ans.  18  ft.  3' 

4.  The  brick  wall  of  a  house  is  108  ft.  3'  high  and 
it  has  an  area  of  9120  ft.  0'  9".     How  long  is  it  ? 

Ans.  84  ft.  3' 

5.  If  the  area  of  a  flower  garden  was  7  ft.  1'  6" 
and  its  length  was  3  ft.,  how  wide  is  it  ? 

Ans.  2  ft,  4'  6". 


384 


INTERMEDIATE  ARITHMETIC. 


6.  If  my  parlor  is  12  ft.  long  and  it  has  an  area  of 
225  ft.  8',  how  wide  is  it  ?  Ans.  18  ft.  9'  8". 

7.  The  area  of  the  ceiling  of  a  schoolroom  is  229  ft. 
11'  2"  10'",  and  the  room  is  12  ft.  6'  4'"  wide.      How 
long  is  it?  Ans.  18  ft.  4'  3". 


CHAPTER  XVI. 

PERCENTAGE. 

LESSON  I. 

366.  PERCENTAGE  or  PER  CENT,  means  by  the 
hundred. 

It  may  be  applied  to  any  thing ;  thus,  6  per  cent, 
of  100  dollars,  4  per  cent,  of  25  bales  of  cotton,  10  per 
cent,  of  100  kegs  of  1'ard,  5  per  cent,  of  250  scholars, 
3  per  cent,  of  1000  books,  etc.,  etc. 

6  per  cent,  of  any  thing  is  6  one  hundredths  of  it. 
It  may  be  multiplied  two  ways.  1st.  As  if  it  were 
a  common  fraction.  Thus : 

6  per  cent,  of  100  dollars. 

OPERATION. 

6KAA 
•IjcJJJ  . 

I00XT" 


PERCENTAGE.  385 

The  above  is  read 

6  one  hundredths  of  100  dollars. 

Cancelling  leaves  $0  for  the  answer. 

Or: 

By  the  decimal  method,  which  is  most  commonly 
used.  6  per  cent,  is  6  one  hundredths,  and  is  writ- 
ten .06. 

By  multiplying  100  dollars  by  .06,  we  get : 

100  X  .06  ±=  600 

Cutting  off  the  number  of  decimals  from  the 
product,  gives  $6.00  for  the  answer. 

(100  per  cent,  of  anything  is  the  number  itself. 
100  percent,  is  equal  to  1  whole  one.) 

36T.  From  what  we  have  just  learned,  we  have 
the  following  rule  for  finding  the  percentage  of  any 
number: 

RULE 

Multiply  the  number  by  the  rate  per  cent.,  written 
as  a  decimal,  and  from  the  product  point  off  as  many 
figures  as  there  are  decimal  figures  in  the  multiplier 
and  multiplicand. 

EXAMPLE. 
1.  What  is  3  percent,  of  $15.25  ? 

15.25  X  .03  =  0.4575 

In  the  above  example  there  are  2  decimal  figures 
in  the  given  number  and   2  in  the  rate  per  cent., 
33 


386  INTERMEDIATE    ARITHMETIC. 

making  4  in  both.  Counting  from  right  to  left,  cut 
off  4  figures  from  the  product.  In  the  above  exam- 
ple there  are  but  4  figures  in  the  product.  Place 
the  period  or  decimal  point  before  the  left-hand 
figure,  as  above,  and  to  the  left  of  thr  decimal  point 
write  the  figure  0.  This  shows  that  there  are  no 
whole  ones.  The  above  answer  is  read :  45  cents 

7  mills  and  5  tenths  of  a  mill ;  or,  45  cents  7  mills 
and  one  half  of  a  mill. 

EXAMPLES. 

1.  Write  decimally  the  following     3  per  cent., 
6  per  cent.,     4  per  cent.,     7  per  cent.,     9  per  cent., 

8  per  cent. 

2.  What  is  3  per  cent,  of  2000  dollars  ?     6  per 
cent.  ?     4  per  cent.  ?     7  per  cent.  ?     9   per  cent.  ? 
8  per  cent  ?  1st.  Ans.  $60. 

3.  Express  in  figures  2i  per  cent.,  4|  per  cent., 
6£  per  cent.,  2J  per  cent.,  4£  per  cent. 

NOTE.— In  the  above  examples  reduce  the  fractional  part  to  a 
decimal,  and  write  the  result  to  the  right  of  the  given  per  cent. 
Thus  4£  per  cent,  is  written  .042.  £  reduced  to  a  decimal,  gives 
2  tenths.  Write  the  2  to  the  right  of  the  figure  4. 

4.  What   is   2£  per  cent,  of  3000   dollars? 
per  cent.  ?     6£  per  cent.  ?     2J  per  cent.  ?    4£  per 
cent.  Last  Ans.  $135. 

5.  Write  decimally  2^  per  cent.,  £  per  cent.,   f 
per  cent.,  £  per  cent. 


PERCENTAGE.  387 

NOTE. — In  the  three  last  examples  the  required  answer  is  less 
than  one  cent,  therefore  we  must  write  two  O's  before  it.  Thus, 
,00£.  Finding  the  value  of  £  of  a  cent,  and  writing  it  after  the 
O's,  gives  .00125.  (£  reduced  to  a  decimal  gives  125.) 

6.  What  is  2-J-  per  cent,  of  3500  dollars  ?     %  per 
cent.  ?     |  per  cent.  ?    -J-  per  cent.?     Last  Am.  $8.75. 

This  sign  <f>  is  the  sign  of  per  cent.  6^  is  read  6 
per  cent. 

7.  A  bought  2500  dollars'  worth  of  molasses,  and 
lost  \  f0  of  it.    How  much  did  he  lose  ?     Am.  $3.12£. 

8.  John  Edwards  bought  450  head  of  cattle  for  20 
dollars  a  head,  and  lost  2f$  on  his  way  to  market. 
How  many  did  he  lose  ?    How  many  had  he  left  ? 
How  much  money  did  he  lose  ?       Last  Am.  $247.50. 

9.  A  collector  collected  $25000  in  a  month,  on 
which  he  was  allowed  %fc.  How  much  did  he  make  ? 

Am.  $62.50. 

10.  A  gentleman  spends  8$  for  clothing,  25$  for 
boarding,  2f0  for  washing,  30$  for  spending-money. 
How  much  per  cent,  of  his  salary  does  he  save  ? 

Am.  35$. 

11.  Suppose  his  salary  to  be  $1500  a  year,  how 
much  does  he  save?  Am.  $525. 


388  INTERMEDIATE    ARITHMETIC. 

CHAPTER    XVII. 

INTEREST. 
LESSON  I. 

368*  Interest  is  money  paid  by  the  borrower  to 
the  lender  for  the  use  of  the  money  lent. 

A  lends  B  $1000  for  1  month,  B  gives  to  A  25 
dollars  for  the  use  of  the  money,  when  the  time  ex- 
pires. 

The  25  dollars  in  the  above  case  is  the  Interest. 

The  $1000  is  the  Principal. 

The  1  month  is  the  Time. 

The  $1000 +  $25 =$1025  is  the  Amount. 

If  a  per  cent,  was  expressed  it  would  be  the 
Rate. 

Interest  is  reckoned  by  the  year,  months  being 
fractional  parts  of  a  year,  and  days  fractional  parts 
of  a  month. 

What  is  the  interest  on  $200  for  2  yrs.  at  6#. 

200  x  2  x. 00 =$24.00^5. 
Or: 

200  x  .06=$! 2.00,  Int.  for  1  year. 

12.00  x  2=$24.00,  Int.  for  2  years. 

EULE. 

Multiply  the  Principal  by  the  Rate  for  the  Time. 
The  rate  must  be  always  expressed  decimally. 


KATES    OP   FOEEIGN   MONEY.  389 

1.  What  is  the  interest  on  $200  for  1   year,  at 

3  %  ?  A?is.  $0. 

2.  What  is  the  interest  on  $200  for  2  years,  at 

4  #?  Ans.  $16. 

3.  What  is  interest  on  $3000  for  2  years,  @  44-  #  ? 

^Iws.  |270. 

4.  What  is  interest  on  $1000  for  3  years,  @  2 

^4«5.  $75. 

5.  What  is  the  interest  on  $3500  for  4  years,  @ 
2J#?  -4w*.  $315. 

0.  What  is  the  interest  on  $4000  for  5  years,  @ 

3J  f0?  Ans.  $625. 

7.  What  is  the  interest  on  $0000  for  G  years,  @ 
2  f0  ?  ^4>w.  $720. 

8.  What  is  the  interest  on  $8000  for  7  years,  @ 
2£  ^  ?  ^4/15.  $1400. 

9.  What  is  the  interest  on  $9000  for  8  years,  @ 
3J$?  -4ns.  $2400. 

10.  What  is  the  interest  on  $10000  for  12  years, 
@  4J  #?  -4w«.  85200. 

309.  RATES  OF  FOREIGN  MONEY  (gold  basis). 

Ounce  of  Sicily $2.40 

£  of  Jamaica 4.84 

Pound  Sterling 4.84 

Real  Plata  of  Spain  (one  bit) 10 

Prussian  Dollar 69 

Bremen  Dollar .78  + 

Roman  Dollar 1.05 

Russian  Dollar  (Rouble) 75 

Dollar  of  Denmark,  Norway  and  Sweden     .     1.05 


390 


INTERMEDIATE   ARITHMETIC. 


37O.  PAPER. 

Folio  Post  .     .    .     .     16  by  21  inches. 
Foolscap  ....        14  by  17      " 
Crown 15  by  20      " 

371.  A  sheet  folded  in  2  leaves  is  called  a  folio. 
A  sheet         "         4  "  "         quarto. 

A  sheet        "         8          "          "        octavo. 
A  sheet         "        12  "  "         12mo. 


THE  END. 


\ 


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